simulating perverse sheaves in modular representation...

Proceedings of Symposia in Pure Mathemtin Volume 56 ( 1994), Part 1

Simulating Perverse Sheaves in Modular Representation Theory

EDWARD CLINE, BRIAN PARSHALL, AND LEONARD SCOTT

For some time we have studied the representation theory of semisimple algebraic groups in characteristic p > 0. Of special interest is a celebrated conjecture of Lusztig [Ll] describing the characters of the simple modules when p is not too small relative to the root system.

A similar conjecture, by Kazhdan and Lusztig [KLl], for the composition factor multiplicities of Verma modules for semisimple complex Lie algebras, has already been proved [BB] and [BK]. The method of proof was to establish a correspondence-actually an equivalence of categories-between a category containing the relevant Lie algebra modules and a category of “perverse sheaves”, where powerful geometric methods had already decided the issue (see Kazhdan and Lusztig [KL], and later treatments by Lusztig and Vogan [LV] and MacPherson [Sp]).

Our recent research has centered on constructing a framework putting the salient features of the characteristic p modules as well as the characteristic 0 Lie algebra modules and perverse sheaves under one algebraic roof. The relevant notion is that of an abstract highest weight category, and the related concept of a quasi-hereditary algebra, as defined and developed by us in [CPSl], [CPS2], [CPS3]. In [PSI the second two authors succeeded in show- ing that the relevant perverse sheaves formed such an abstract highest weight category, and that their derived category coincided with the relative derived category of constructible sheaves appearing in the arguments of MacPherson cited above.

Of course, our framework also encompasses the characteristic p modules we wish to study, so now it is possible and interesting to take a different viewpoint: What properties of perverse sheaves can be proved in the

1991 Mathematics Subject Classi’cation. Primary 20G15, 20G05. Research supported in part by NSF Group Project Grant DMS-890-266 1. This paper is in final form, and no version of it will be submitted for publication elsewhere.

0 1994 American Mathematical society 0082-0717/94 $1.00 + $.25 per page

63

64 EDWARD CLINE, BRIAN PARSHALL, AND LEONARD SCOTT

framework of a general highest weight category? What ingredients are necessary to reproduce the geometric arguments of MacPherson?

In this paper and a previous one [CPS4] we investigate these questions. Originally these papers were combined under the title of [CPS4]. Roughly speaking, the main results of the original manuscript, reviewed below and in $2, are proved in [CPS4], while the present paper includes further details and issues more strictly concerned with the extent to which general highest weight categories imitate perverse sheaf categories. For example, we give an algebraic version of the Deligne-MacPherson characterization of perverse sheaves in (1. l), (1.2). A t-structure and algebraic cohomology groups cap- turing the ordinary cohomology groups of perverse sheaves are discussed in $3. We also review in (2.4) the “parity theorem” of [CPS4, $43, used there as a weak replacement for Gabber’s purity theorem [BBD], a key instrument in MacPherson’s arguments. With a suitable parity condition, axiomatized in [CPS4] as the “existence of a Kazhdan-Lusztig theory”, MacPherson’s perverse sheaf arguments can be made to work in our context; this is carried out in §§4 and 5.

This development, treated in abbreviated form in [CPS4], resulted in a number of reductions of the Lusztig conjecture. Also, the “enriched Grothen- dieck group”, required as a substrate for the argument, led to complete calcu- lations of all Ext” groups between irreducible modules under the “Kazhdan- Lusztig” theory hypothesis, in terms of “Kazhdan-Lusztig polynomials”.

In somewhat more detail, let G be a semisimple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p . For a dominant weight 1, let L(I) be the corresponding irreducible rational G-module of high weight ;1. A main result of [CPS4, (5.4)], establishes that the Lusztig conjecture is equivalent to the simple assertion that

(O-1) Ext;W), W’)) # 0

for p-regular dominant weights 1 and rZ’ which are mirror images of one another in adjacent p-alcoves and which satisfy the Jantzen condition; see

(2.6) below. (In particular, Iz and L’ lie in the lowest p2-alcove. Using the translation principle, one can assume that I and A’ lie in the orbit of 0 under the “dot” action of the affine Weyl group.) Section 5 reviews other equivalent forms of this reduction from [CPS4].

Conditions like (0.1) emerged from our study of persexves sheaves, and a related vanishing condition forms the hypothesis of the “parity theorem” (2.4). Interpreted for complex Lie algebras, our development settles an old question as to whether “even-odd” vanishing of certain Ext”-groups formally implies the “Vogan conjecture”. (See (5. lOc).) We also mention that the calculation of Ext”-groups mentioned above applies in the case of the category B to yield a complete determination of the groups Ext$(L(A) , L(r)) for all n and all integral weights ;Z, r . (See [CPS4, (3X2)].)

SIMULATING PERVERSE SHEAVES 65

1. Preliminaries, and an abstract version of the Deligne-MacPherson characterization of simple perverse sheaves

There is a well-known characterization due to Deligne and to MacPherson of the simple perverse sheaves in terms of conditions on the support of their cohomology groups. (Cf. [BBD, (2.1.9)] and [BK, p. 4051.) In this section, we give a version of the Deligne-MacPherson characterization of simple perverse sheaves valid for certain abstract highest weight categories. This characterization will be in terms of “induced” and “Weyl” modules. In the framework of perverse sheaves on a stratified complex manifold X , the corresponding “induced modules” typically have the form Ri,, (C[dim S]) and the “Weyl modules” have the form Li,,(C[dim S]) = i,!(C[dim S]) for the inclusion map i, : S + X of a stratum S (which is contractible in standard examples). The simple objects are shifts (by dims) of intersection cohomology sheaves associated to the strata.

The characterization itself is less important in our theory of Hecke operators, as developed in 35 of this paper. We present it mainly as a first illustration of a meaningful perverse sheaf/highest weight category translation. However, the techniques used in the proof are important, and they will reappear later.

Before stating our characterization, we introduce some of the algebraic highest weight categories of interest, and we establish some conventions used throughout this paper.

(1.1) Notation, conventions, and examples. Fix a field k , and let G?? be a highest weight category over k having (interval finite) weight poset A. We will closely follow the conventions made in [CPS4] (which are slightly different from those of [CPSl] where the notion of a highest weight category was first formally introduced). Thus, we assume that each object in g has finite length. The poset A indexes the irreducible ‘Z-objects, up to isomorphism: if 1 E A, let L(rZ) be the corresponding irreducible. We assume that for L E A, End&L(n)) g k , and that g has both “induced objects” A(1) and “Weyl objects” V(n) , A E A. These satisfy the usual properties listed in [CPSl, (3.1)]. (Thus, the V(n) are the induced objects in the highest category go’ dual to %?‘) . It follows that ‘Z has both enough injective and enough projective objects. As discussed in [CPSl, $31, when the weight poset A is finite, the existence of the Weyl objects follows automatically from the definition of a highest weight category given there.

In the following, we give several important examples of highest weight categories. We also include some further notation we will use throughout the paper.

(1.1.1). Let A be a finite dimensional k-algebra whose radical quotient A/ rad(A) is k-split. The corresponding category %? = mod A of finite dimensional right A-modules is a highest weight category (relative to some


poset structure on the finite set A of isomorphism classes of simple modules) if and only if A is a quasi-hereditary algebra as defined in [CPSl, 531. (This basic result is proved in [CPSl, (341.) Conversely, if ‘Z is a highest weight category with finite weight poset, then ‘Z is equivalent to a module category mod A for a finite dimensional algebra A with A/ rad( A) k-split. (The algebra A is determined only up to Morita equivalence, and, by above, is quasi-hereditary.) In the sequel, we will often use the fact that, if @? has a finite weight poset, then 9 has the Krull-Schmidt property. This is clear from the realization of %’ as a module category for a finite dimensional algebra.

Recall that a duality on %’ is a contravariant equivalence D: %? -+ %?

satisfying DL(1) g L(1), Iz E A, and D2 g id, . As a first example of a highest weight category, let g be a complex semisim-

ple Lie algebra, and fix a Cartan subalgebra g and Bore1 subalgebra b containing b . Consider the corresponding category B of [BGG]. This is well- known to be a highest weight category, although it cannot be of the form mod A (for a finite dimensional algebra A) because it does not have finitely many simple objects. However, fix a weight 3, E g* , and let 8’ be the full subcategory of B consisting of objects having composition factors L(y) with high weights of the form y = w (2 + p) - p . Here p E b* denotes the sum of the fundamental dominant weights (relative to 6) . Then @i is a highest weight category of the form mod A for a finite dimensional quasi-hereditary algebra A . This categoory admits a duality.

For the reductive group G = GL, over k , the closely related theory

of the Schur algebras S(n , r) = End, (I’@) [Gr] provides important ex-

amples of quasi-hereditary algebras and therefore highest weight categories %7(n) r) = modS(n, r) [PI. (Here the symmetric group 6, of degree r acts

by permutation of factors on the space V@” of r-fold tensors associated to a vector space V of dimension n .) The weight poset of g(n , r) identifies with the set h+(n) r) of partitions of r into at most n parts. (These partitions can be written 1= (A, , . . . , A,) where A1 2 . . . 2 1, 2 0 and C Ai = r .

They are partially ordered by putting rZ 5 1’ if and only if CF li 5 x: d: forall kin.)

( 1.1.2). Let G be a semisimple, simply connected algebraic group over an algebraically closed field k of positive characteristic p . Assume that G is defined and split over Fp . Let T be a fixed %-split torus, and denote the

root system of T acting on the Lie algebra of G by Q, . We choose a set @’ of positive roots, and let B denote the Bore1 subgroup corresponding to the associated set Q- of negative roots. The set X(T) of characters on T is partially ordered by the rule: I 5 p * p - il = CaEa+ naa: for nonnegative

integers na . We also have an induced poset structure on the set X(T)+ of

dominant weights (relative to Q’) . (A second poset structure on X(T) and X(T)+ , defined by strong linkage, will be discussed in ( 1.1.5).)


Denote the Frobenius morphism on G by I; : G + G, and let q be the category of finite dimensional rational G, - T-modules for the group scheme

G, s T = F-‘(T) . The irreducible G, . T-modules are indexed by the poset X(T) , and q is a highest weight category with duality. For more details, see [PS, @I.

( 1.1.3). For an ideal l? in the weight poset A of a highest weight category ‘8, the full subcategory @?[I] of $9 consisting of objects having composition factors of the form L(y), y E I, is a highest weight category having weight poset I’. The natural full embedding functor i, : %?[I] + ‘$7 maps the A(y) , V(y) E Ob(%?‘[I]) to the corresponding objects in 55’.

If Sz c I is a finite coideal, the quotient category ‘ZY(LI) = ‘Z[I]/5?Y[I\n] is a highest weight category with weight poset 0. The quotient functor j* : szp-1 + ziqt2) sends A(L), V(n), L(A), ,? E L2, to the corresponding induced, Weyl, and simple objects, respectively, in @Y(R) [CPSZ; P, $11. For example, consider Schur algebras (1.1.1). For integers N > n > 0, A+(n) r) identifies with a coideul in h+(N, r) , and @?(N, r)(A’(n , r)) E E(iV, r)/E(N, r)[A+(N, r)\[A+(n, r)] identifies with @Z(n) r) . (A proof can be based on [Gr, $61.) Therefore, j* : @(ZV, r) + @(n , r) relates the representation theory of GL, with that of GL, . (Relative to (1.1.6) below, similar considerations apply to the q-Schur algebras [DJ], associated to the quantum group GL4(n) [PW, $61. These results apply even if k is replaced

by the ring Z[q112, qm1j2] [CPS3].)

(1.1.4). Let G be as in (1.1.2). Let 55 (resp., 57’) denote the abelian category of finite dimensional (resp., arbitrary dimensional) rational G-modules. Then ‘@ is a highest weight category in the sense of [CPSl], but not in the present sense, since it has objects of infinite length. On the other hand, %? does not contain enough injective objects, so it is not a highest weight category. However, let I be a finite ideal in the poset X(T)+ . The full subcategory %:[r] (defined as in (1.1.3)) is a highest weight category. Also,

A(I) is the induced module indz 1 while V(L) is the usual Weyl module of highest weight 1. Also, 5?[I] has a natural duality [CPS2, (3.3)].

(1.1.5). Continuing as in (1.1.4), for (Y E @, rz E Z,

s a,n,:X(T)@aR+X(T)@R, x H x - ((x ) a”) - np)a

defines an affine reflection. These reflections generate a Coxeter group Wp

(the “affine Weyl group” of G) . If II c cP+ is the set of simple roots, and if a0 E a’ is the maximal short root, the set Z = {so a 1 (Y E II} U {sa p}

defines a set of simple reflections (i.e., ( Wp , Z) is a Coxeter system). Be&es the usual action of Wp on X(T) 8 R , we also use the “dot” action defined by w.x=w(x+p)--p, XEX(T)@R, w E Wp.

We will also use another partial ordering t on X(T)+ - p , where p =


$ C&*+ a * For 1, v E X(T) , write A 1 u provided there is a sequence

(1.1.5.1)

in X(T)+ - p such that, for 0 I i < t , there exists a positive root ai and a positive integer ni with se, np . Ai = Ai+r . The partial ordering is discussed

in [Al, Proposition 11. A’ ‘similar partial ordering defined on the alcoves contained in X(T)+ - p may be viewed as a partial ordering on the set of distinguished right coset representatives of W in Wp . The definitions of these partial orderings on weights and alcoves differ from those in [J, $11.6.51, although Verma [Ve, 3 1.61 asserts (without proof) that the latter ordering on alcoves contained in X(T)+ - p agrees with the Bruhat ordering, and thus with the above ordering. More recently, a proof of this fact has been given by Ye [Y] (using a case-by-case argument) and by Wang [W] (using an inductive argument).

Let I be a finite ideal in (X(T)+, r) . By [CPS4, 551, the full subcategory ‘8[r] of all finite dimensional rational G-modules having composition factors L(y) , y E l7, is a highest weight category (having a duality) with weight poset (I, t) . Also, the objects A(1) and L’(n) coincide with those defined in (1.1.4).

Fixaweight 1,andput O,‘={~~x(T)+lv=w.iZ forsome WEW~},

regarded as a poset by restriction of t. Let gi denote the full subcategory of the category ‘Z of finite dimensional rational G-modules whose objects have composition factors L(r) , r E 0: . Let I? be a nonempty, finite ideal

in 0:. Then the full subcategory gA[r] of %A consisting of rational G- modules having composition factors L(y) and y E I is a highest weight category relative to the poset (r , 1) .

(1.1.6). Let %‘q be the category of finite dimensional rational modules for

the quantum group G, = GL,(n) over an arbitrary field k . For simplicity assume that the parameter q is a primitive &h root of unity for some odd integer L > 1 . If k has characteristic zero, %q is a highest weight category

[PW, (9.10.5), (11.5.2)]. When k has positive characteristic, the correct highest weight categories have the form gq[[r] for some finite ideal I of dominant weights. Using the linkage principle of [PW, (10.3.5)], one can generalize the results ( 1.1.5); here the affine Weyl group Wp is replaced by the

analogously defined affine Weyl group W, , and we put 0: = W, -1 n X(G)+ . Similar remarks apply to the quantum special linear group SL,(n) , using

Fw This example generalizes to all types, by working with quantum enveloping

algebras (and type 1 representations). As in [APW], let U, be a quantum enveloping algebra over F = Q[o] , where o is a primitive &h root of unity for an integer e which is a power of an odd prime > 3. The irreducible finite dimensional type 1 representations are indexed by the poset X+ of dominant weights of the associated root system. Let gt denote the category

SIMULATINGPERVERSESHEAVES 69

of finite dimensional type 1 representations for U, . As in (1.1.4), using results in [APW], we obtain a highest weight category gi;[I] for any finite ideal I of weights. Also, by the linkage principle for quantum enveloping algebras [APW, (8. l)], the results ( 1.1.5) extend to the quantum enveloping case.

It is possible to consider U, over fields k other than F , although current technology requires further restrictions on the base field. However, in the GL,(n) case, no such restrictions are required. Work of Dipper and James (see [Di]) ties its representation theory into the representation theory of the finite groups GL(n , F,) in nondescribing characteristics. One would hope this point of view could be extended to all types (as emphasized by us in talks at the November, 1990 MSRI conference).

( 1.1.7). The perverse sheaf categories, discussed briefly above, provided examples of highest weight categories. More details can be found in [I%, $51. Our only use of perverse sheaf theory in this paper is in the way of analogy.

Let ‘Z’ be a highest weight category, regarded as fully embedded in the

derived category Db (%‘) of bounded complexes in %? by sending any object in ‘%’ to the complex concentrated entirely in degree 0. The result below gives

a homological characterization of @F c ob(%?) in terms of the induced and Weyl modules. This result is a perfect analog of the geometric description of the abelian category of perverse sheaves given in [BBD]. (See $3 for a way to recover the cohomology groups of a perverse sheaf in an abstract setting.)

(1.2) THEOREM. Let g be a highest weight category with weight poset A. An object X in Db(ST) is isomorphic to an object in ‘%? if and only if the following two conditions are satis-ed for all 1 E A :

(1) HOm;bcg, (JW),X)#O*nHk

(2) HOm;bcgj (X,A(I))#O*n>O.

PROOF. It is clear that any object X in ob(5?) which is isomorphic to an object in %Y satisfies the conditions (1) and (2) stated in the theorem.

Conversely, suppose X E Ob(ob(E)) satisfies the two conditions. To show that X is isomorphic to an object in 5F, it is sufficient by a standard argument [BBD, (1.3.2)] to prove that H”(X) = 0 for n # 0. If H”(X) # 0 for some n < 0, let m be the smallest such integer n . By a truncation argument, we can assume that X is a complex concentrated in degrees m , m + 1, . . . . Let 1 E A be such that L(Iz) occurs in the socle

of Hm(X) 2 Ker(X” + Xm+‘) . It follows that Hornzb(@)( V(n), X) # 0,

contradicting condition (1). Thus, X has cohomology concentrated entirely in nonnegative degrees. A dual argument, now using condition (2), shows that X has cohomology concentrated entirely in nonnegative degrees. 0

Next, we give a result which, together with (1.2), is parallel to the Deligne- MacPherson characterization of simple perverse sheaves.

70 EDWARDCLINE,BRIANPARSHALL,ANDLEONARDSCO'l-I

(1.3) THEOREM. Let X E Ob(‘Z) f or a highest weight category 59. Then X S’ L(I) for some weight 1 E A if and only if the following three conditions hold:

(1) Homg(X, A(v)) = 0, Vv E A, v # 1; (2) Hom&(V(v), X) = 0, Vu E A, u #J.; (3) Horn& v(n), X) % k .

PROOF. Trivially, (l)-(3) hold for X % L(A) . Conversely, suppose the stated conditions hold for X E Ob(@) . Let r E A be maximal such that L(z) is a composition factor of X . Let F be the ideal in A generated by all weights y such that L(y) is a composition factor of X . Then X ,

v(7) E OWWI) and, in fact, since 7 is maximal in F , V(7) is projective in ‘Z’[F] . Hence there is a nonzero homomorphism V(7) + X , so that (2) implies that 7 = 3, . Thus, all the composition factors L(V) of X satisfy v 5 1. Clearly, (1) and (2) imply the simple factors of the socle and radical quotient of X are all isomorphic to L(A). Finally, the projectivity of v(I) in %‘[F] and condition (3) immediately give that X Z L(1). 0

2. A review of Kazhdan-Lusztig theory

This section gives a brief exposition of some results from [CPS4]. As motivation, we begin with remarks concerning the category B (see (1.1.1)). Following the notation of [KLl], the simple modules in the principal block are indexed by the elements of the Weyl group W : thus, for w E W, we put L, = L(-wp - p) , the irreducible module of high weight w . (-2~) = -w p - p in the category B . Similarly, let M, = V( -w p - p) denote the Verma module of high weight -wp - p . The character of L, is given in terms of the characters of the Vet-ma modules by the formula

(2.0.1) ch(L,) = ~(-l)e(w)-ICv)~,,u(l)ch(My).

This result, originally known as the (classical) Kazhdan-Lusztig conjecture, was conjectured in [KLl], and proved by Brylinski-Kashiwara [BK] and, independently, by Beilinson-Bernstein [BB]. Here the Py w are the Kazhdan-

Lusztig polynomials in q = t2 associated to W [KLl]. Work of Vogan [Vl], based on the Kazhdan-Lusztig conjecture, implies the polynomials Py ‘u) have the representation-theoretic interpretation

(2.0.2a) Py , 2u (q) = c dim, ExGw)-‘(y)-2i(My , LW)qi .

This is, in some sense, a special case of a very general Euler characteristic formula of Delorme. Delorme’s formula involves a polynomial whose definition is very similar to the Vogan interpretation above. However, the remarkable feature of (2.0.2a) is that the expression for Py w involves only

even powers of t . Explicitly, Vogan shows that

(2.0.2b) Extp)-L(Y)-” (M,,L,)#O+n-0 (mod2).


By [CPS4, $31, such a property allows a calculation of all Ext” groups between simple objects in a very general setting, namely that of a highest weight category having a duality and a function e : A + Z (a “length” function).

In this case, the bounded derived category Db(!9) contains two natural full

subcategories & and gR. These are defined precisely below, but roughly

speaking, JYL (resp., sR) consists of complexes having “filtrations” with sections of the form I’(n)[s] (resp., A(l)[s]) , and 1 E A and s = e(J) mod2. The main point is that the natural representation-theoretic polynomials defined below in (2.0.5) associated with a simple object L(1) involve only even

powers of t if and only if L(J)[-f?(n)] belongs to &.

Equivalently, in the presence of the duality, L(I)[-C (A)] belongs to ZYL n fYR. This latter property is taken in Definition 2.1 as characteristic of an abstract Kazhdan-Lusztig theory. Thus, from our point of view, the existence of a Kazhdan-Lusztig theory is a matter quite different from the calculation of any Kazhdan-Lusztig polynomials. The condition on L(L) [ -e (A)] , however, guarantees that L(1) is represented in two different Grothendieck groups (see below). In these representations, the coefficients of the canonical dual

bases provide natural polynomials p,” rl and p,” 1 associated to L(1) . These coincide with the Poincare polynomials defined ‘more generally in (2.0.3) and (2.0.4). Of course, in our abstract setting, the polynomials are indexed by weights rather than elements of a Coxeter group. In the case of the principle block in the category d , we work with weights u , I which lie in W - 0 = We (-2~). Write ;Z = 20. (-2~) and Y =y.(-2~) for y, w E W. Then the “Kazhdan-Lusztig” polynomial P, A defined in (2.0.5) below identifies with the Kazhdan-Lusztig polynomial ‘P,, ~ as defined in [KLl], once it is

converted into a polynomial in q via the substitution q = t2. The (time- honored) convention of indexing the polynomials relative to -2p is natural, in our view, for the category B since -2p is minimal in its orbit, forcing I/(-2~) Z L(-2~) . This concludes the motivation.

For the remainder of this section, fix a highest weight category ‘$7 having

weight poset A and a length function C : A + Z. If X E Ob(Db(%)) , we define, for each v E A, associated left and right Poincare “polynomials”

P,L,X) p,R,x E Z[t , t-l] by the formulas

(2.0.3) d,X =c dim, Hom$p)(X, A(u))t” and

n

(2.0.4) &=C dim, Homia(g)(V(v), X)t” . n

It is easily verified that p,“, X and p,“, X belong to Z[t , t-l] [CPS4, 531. Of

course, if X E Ob(‘Z) , then, for all v E A, p,” x , p,“, x E Z[t] .

The polynomials p,“, I = p,“, L(1) and p,” I E if LCnj are the Poincark polynomials of the simple module L(1) . The kazhdakLusztig polynomials P,, 1


are given by the formula

(2.0.5) P” * E pf 1 = p-+y I.

These polynomials (2.0.5) are indexed by pairs of weights in A, while the classical Kazhdan-Lusztig polynomials [KU] are indexed by pairs of elements in a Coxeter group. The relationship between these two notions, one homological and one combinatorial, will be discussed in detail in $5 (see also

(2.8)). The notation P(t) means P(t-‘) .

We now define the full subcategory gL of Db(‘@?) to consist of all objects isomorphic to an object of the form X, * . . . * X, , where Xi = V(n,)[s,] for some lj E A and some si = !(A,) mod 2. Here we are using the *-operator

of [BBD, 1.3.91. Dually, the full subcategory 8” is defined using the A(I)

in place of the V(n). Set gL = EL @ 8’[1] (resp., gR = gR @ sR[l]).

An important characterization (the “Recognition Theorem”) of gL is es-

tablished in [CPS4, (2.4)]: Assume that A is Jinite. Then X E Ob(Db($!7))

belongs to gL if and only if for all integers n and all I E A, we have Horn” (X , A(I)) # 0 + n - e(J) mod 2 . A similar result holds for gR . When A is not finite, it is best to take the above “even-odd” vanishing property as the definition of gL . In this paper, as in [CPS4], we are concerned mostly with the finite case; see [CPSS] for a treatment of the infinite case.

Associated to gL, there is a left “enriched” Grothendieck group Kt (B, 1) . Namely, we take the free abelian group on symbols [Xi * . . . * X,] modulo relations [Xi * . . . * X,] = [Xi] + . .. + [X,] , where X, * . . . * X,, is as above

(with an evident modification if X, * . f . * X, belongs to ZL[ 11) . As proved

in [CPS& 523, Ki(B , C) is a free Z[t , t-‘l-module with basis consisting of

[V(J)], I E A. Dually, a right “enriched” Grothendieck group Kf(B , !)

is defined; it is a free Z[t , t-‘l-module with basis [A(n)], 1 E A. An important result [CPS4, (2.3)] is the existence of a nondegenerate sesquilinear pairing

(2.0.6) ( ) ): K,L(S, 1) x K,R(B) e> + z[t, t-l]

in which the above bases are dual. If X E Ob(@‘) , then we have the expression

(2.0.7) WI = ~&Jw41. UEh

Ah P,” x = ([Xl, [A(v > hold for ‘Y E Ob(gR) .

using the pairing (2.0.6). Similar expressions

For X E Oh(g) , let ch(X) be the image of X in the Grothendieck group K,(g). As discussed in [CPS4, (3.2)],

(2.0.8) WX) = ~&WI ch v(v) = I& chA(v) VEA VEA

for the “character” of the object X .


We now give the definition of a Kazhdan-Lusztig theory for a highest weight category having a finite weight poset A. For the more general sit- uation, see [CPS4], [CPSS].

(2.1) DEFINITION. Let @? be a highest weight category with length function e : A + Z . Then 9 has a Kazhdan-Lusztig theory with respect to C provided that for each A E A,

L(A)[-l(A)] E Ob(& n gR).

(2.2) REMARK. Let @? be a highest weight category. If X lies in @‘,

any direct summand of X does also. Since Db(‘5?) is well known to be a “Krull-Schmidt category” [H] (i.e., every object is expressible as a direct sum of indecomposable objects, uniquely up to isomorphism), it follows that

@’ is a Krull-Schmidt category. So are gR and @’ n gR by the same

argument. It follows that the pair gL, 8’[1]) is a “torsion pair” in the

sense of [HI. This means that Hom,b(V,)(X, Y) = 0 for X E Ob(gL) and

Y E Ob(g’[ 11) , and any X satisfying these vanishing properties for all

Y E Ob(sR[l]) necessarily belongs to the subcategory &?. Also, we see that if X + Y + Z + is a distinguished triangle in which X , Y , Z E Ob(@ n sR), then Y z X @ Z . Finally, it is worth noting that, if the sim-

ple object L(A) belongs to @’ (resp., 2’“)) then necessarily L(A)[-!(A)] E

Ob(&) (resp., L(A)[-e(A)] E Ob(8’)). For example, if L(1) E Ob(@‘) , then the fact that End(L(A)) z k clearly implies that L(I) belongs to ei-

ther gL or to &[ l] . However, since there is a nonzero homomorphism

L(A)[-e(A)] + A(J)[-!(A)] and A(n)[--e(n)] E Ob(gR), it must follow that

L(A)[-l(A)] E Ob(&) , as desired. (2.3) REMARKS. (a) Let A be a finite dimensional algebra of global dimen-

sion 5 2 such that A/rad(A) is k-split. Then A is quasi-hereditary [DR], so that @? = mod A is a highest weight category as explained in ( 1.1.3). It would be interesting to determine when 5% admits a Kazhdan-Lusztig theory relative to some length function. Suppose that A is hereditary (i.e., gl . dim(A) 5 1) and, for simplicity, that the weight poset A is such that, for each weight 1, we have that A(1) = L(1) and that V(n) is the projective cover of L(A) . (This is always possible.) It is then easy to determine when 5? has a Kazhdan-Lusztig theory. In fact, we can replace A by a Morita equivalent algebra to assume that it is basic. Then A is isomorphic to the path algebra ki for a directed graph A having no oriented cycles (see [HI). We say that two simple modules L(1) and L(v) have “opposite parity” pro-

vided ExtL(L(1), L(v)) # 0. Clearly, this relation is consistent if and only

if each (necessarily unoriented) cycle in i has an even number of edges. If this holds, we can define a length function e compatible with the parity, and g has a Kazhdan-Lusztig theory.

(b) Consider a finite poset A, and let A be the associated poset algebra, as discussed, for example, in [PS, $61. Then A is a quasi-hereditary algebra

74 EDWARDCLINE,BRlANPziRSHALL,ANDLEONARDSCOTT

with weight poset A. In the highest weight category %’ = mod A , the Weyl module V(n) identifies with the projective indecomposable cover P(A) of the simple module L(1) . Also, we have A(A) = L(I) for all weights A. Suppose that .! is a length function defined on A. We use e to assign a “parity” to each simple module L(1), according to whether a(n) is even or odd. A necessary condition that $9 admit a Kazhdan-Lusztig theory is

that Extb(L(n) , L(V)) # 0 imply that L.(A) and L(V) have opposite parity. It is very easy to write down simple examples of posets in which such an assignment of parity is not possible. (For example, the “pentagonal” poset A = {a, b, c, d, e} defined by a > b > e and a > c > d > e has this property.) It would be interesting to have a combinatorial characterization of those posets A for which the associated highest weight category does admit a Kazhdan-Lusztig theory (relative to an appropriate length function).

Another basic result is the following parity theorem, which guarantees that certain objects in a highest weight category are completely reducible.

(2.4) THEOREM [CPS4, (4.1)]. Let g be a highest weight category having weight poset A, andJix X E Ob(%) . For 1, u E A, assume that

Horn&X, A(1)) # 0 # Horn&X, A(v)) + Extb(L(v), A(1)) = 0,

and

Hom&V), X) # 0 # Horn&V(v), X) + Extb(V(I)L(u)) = 0.

Then X is completely reducible. q

We conclude this section by briefly indicating a main application of the theory of this section to the modular representation theory of semisimple algebraic groups. Thus, let G be as in (1.1.4). Assume that p 2 h , the Coxeter number of G. Fix a dominant weight 1 lying in the bottom p- alcove, i.e., 0 < (A + p, a”) < p for all positive roots cr. Let F(n) be the ideal in (0: , 1) consisting of all weights r which satisfy the Junzten condition

(t+p,Q;)sp(P-h+% where ‘~a is the maximal short root. Recall that the Lusztig conjecture [Ll,

531 asserts, for w . L E r(n), that

(2.5) chL(w .A) = C-1) w-wp

ywo,wwoW) ch J’(y -A) - yw,gmJ, ,y-&X(T)+

Here the P,, v , u , v E Wp , are the Kazhdan-Lusztig polynomials [KM] for

Wp , and w,, E IV is the long word. We regard P,, ‘u(q) = P,, ‘u ( t2) as a polynomial in t . Then we have the following result. ‘(In the context of the results of this paper, 55 contains further discussion of the proof of this result; see (5.9).)

(2.6) THEOREM [CPS4, (5.3)]. Let G be a semisimple, simply connected algebraic group over k . Assume that p 2 h, the Coxeter number of G,


and fix 1 E X(T)+ in the bottom p-alcove. The following statements are equivalent:

(a) The highest weight category ‘S?[r(A)] has a Kazhdan-Lusztig theory relative to the length function ! on (r(A) , t) defined by e(w . A) = a(w) .

(b) The character formula (2.5) is valid whenever w -1 E r(A) . (c) Whenever z, o E T(A) lie in adjacent p-alcoves, then Ex&L(r) , L(a))

#O. Zf these conditions hold, the Kazhdan-Lusztig polynomials (2.0.3) identi’

with the Kazhdan-Lusztig polynomials for Wp as follows:

P y*L,w-l =puJy ww 0’0 (W.~,y-hr(q).

As briefly noted in [CPS4], the above theorem holds in a wide variety of situations, e.g., the category 6’) and various categories for quantum groups and enveloping algebras. We will return to these questions in $5.

3. Cohomology

Let @? be a highest weight category with weight poset A. In this section, we assume that A isfinite. As a guiding principle, we think of g as a suitable category of “perverse” sheaves in the sense of [BBD]. Pursuing this analogy, the idea behind the following result and the next is to “go backwards” from the perverse sheaf category to the original category of constructible sheaves.

We are even able to describe functors ‘Hn , n 2 0, on arbitrary highest weight categories (with a length function C) which, for a perverse sheaf, give its (constructible) cohomology as a complex of sheaves. (One should think of the function L below as the negative of a perversity used in forming 59

as a category of perverse sheaves.) We use these functors eZZ” to give, in (3.3), a “geometric interpretation” of the recognition theorem [CPS4, (2.4)].

For the notion of a t-structure on a triangulated category, see [BBD, $1.31. The construction described below, using t-structures and perversities, was first considered in [PS, $51, using the notion of a “normality” as defined there.

(3.1) PROPOSITION. Let ‘%’ be a highest weight category with finite weight poset A and length function !? : A e.9’o of Db(%Y) by setting

+ Z. Define full subcategories e9’Q and

and

Obtg”) = {X E Ob(Db@)) 1 Hom$cgJ(X, A(v)) = 0,

VU E A, Vn < -C(u)},

Ob(L&o){X E Ob(Db(a)) 1 Hom$(g)(Y(y), X) = 0, VV E A, Vn < e(u)}.

Then the pair (e.CB’o, 5 e92o) defines a nondegenerate t-structure on Db(‘%Y).


PROOF. We first argue by induction on the cardinality (Al that (e~‘o,

l&O) defines a t-structure on Db(F) . If A = {A}, it is easily seen that

(eL$5o, ‘.@‘) is the shifted t-structure (Db(!Qy(‘) , Db(‘@Le(n)) of [BBD,

(1.3.2(i))], i.e., Db(%F)‘c(n) (resp., Db(@Y)2L(n)) is the full subcategory of

ob(‘Z) consisting of complexes X such that H”(X) = 0 for n > e(J) (resp., n < c(n)) .

So suppose that the theorem holds for all highest weight categories having posets of smaller cardinality than [A[ , and let 3, E A be maximal. Set r = A\(n). By [CPSl, $31, there is a recollement diagram

.* I 2

(3.1.1) D’(SC??[r]) i Db(g) < Db(Vect,) !

4L &

(see [BBD, 3 1.4.31). H ere we identify the quotient category Db(F)/~b(~[~)

2 Db(%F(l)) with Db(Vect,) using the natural isomorphism @Y(L) E g:/g[I]

Z Vect, . We may assume, by induction, that (e’rs’o, ‘lr&‘) defines a

t-structure on Db(%[r]) . We assign to Db(Vect,) the shifted t-structure

PbWectk)- , ( <‘(‘) Db Vectk)“(‘)) . Using [BBD, (1.4. lo)] and (3.1. l), these

t-structures induce a t-structure @“, %‘O) on Db(E’) defined by

%” = {X E Ob(Db(S)) 1 j*X E Ob(Db(Vect,)‘e(“)) and i*X E Ob(L”g”)}

and

%” = {X E Ob(Db(%Y)) 1 j*X E Ob(Db(Vectk)2L(‘)) and i!X E Ob(e’r&o)}.

(See [BBD, 51.4.91.) We claim the pair (%‘O, %‘O) coincides with the

pair (LL8”, ’ .&‘) of subcategories defined in the statement of (3.1). To see this, observe that the functor i, preserves the objects A(y) and V(y) for weights y in r. (More precisely, if A,(y) and I$(y) are the induced and Weyl objects in E[I’J, respectively, corresponding to y E I, then i&(y) G A,(y) G A(y) and i,V,(y) 2 VA(y) E V(y) .) Hence, if

X E Ob(ob(E)), we have Homib(orrl,(i*X, A,(y)) E Homia(,)(X, A(y))

and HomLa(glr.l)( I$( y) , i!X) E Homn Dbcgj (V(y) , X) for all n and all y E r . We observe next that j,k E A(I) and j,k E v(n). This follows because

j* (resp., j,) is the right (resp., left) derived functor of an exact quotient

functor ‘Z + Vect, E’ g’(n) inducing j* = j! ; see [CPS3, 533. Hence we have the following isomorphisms for all n :

Hom;b(vmtk,(j*x > k) = HOm;b(&X, A(A)) ;

Hom&retik)(k, j*x) g Hom;b(g)(V(l), x).

Now it is straightforward to verify that ‘L%” (resp., ‘&‘) coincides

with .%‘O (resp., %‘O) , establishing the claim above. Thus, (Lg’o, eL8’o)

defines a t-structure on Db(@) .


To verify that the t-structure is nondegenerate, we may assume, by in-

duction, that the t-structure on Db(‘Z[r]) defined by Cl, is nondegenerate.

Therefore, since the t-structure induced on Db(Vect,) is obviously nonde-

generate, the nondegeneracy of the t-structure on Db(5?) follows from [BBD, (1.4.11)]. 0

We may thus apply the theory of t-structures developed in [BBD]. We

let e2’ E e9’” n e.9Lo be the corresponding heart of the t-structure. It

is a full abelian subcategory of Db(SC). Also, let ‘rCo : Db(5?) + e~‘”

and er,o : Db(%?) + e.9’” be the corresponding truncation operators. Put

eH” = 7T<o 0 eT>o : D’(E) -t e2’, and set eH”(X) = eH”(X[m]) for all - objects X in D’(S) and all integers m . Finally, recall that we have,, for

an arbitrary integer m, subcategories e9’” = e9’“[-m] and e~‘” =

e.&“[-m]. If D is a duality on a highest weight category 527, it extends to an ex-

act, contravariant functor D : Db(E7) + Db(‘Z) such that D2 g id, and

H”(DX) 2 DH-“(X) for all integers n and all objects X in Db(E) .

(3.2) PROPOSITION. Let E be a highest weight category having a Jinite poset A. Let ! : A + Z be a length function. Then the following statements hold

(a) Zf D is a duality on ‘8?‘, D(eg’o) = -e~‘o, D(e&o) = -eg’o, and D(eSY) = -eGY. Also, D o er<o g -e~,o o D and D o e~<O S -et<o o D. Finally, for any integer m , we have -

- -

(b) Assume that e satisjies the condition: A 5 u + e(n) 5 e(u). Then V(n)[-e(n)] E Ob(eX) VA. Further, e%’ is a yinite) highest weight category with weight poset A and WeyZ objects {e V(V) E V(v)[-e(v)] ( v E A}.

(c) Assume that the length function t? is compatible with the poset structure on A (i.e., 15 u + k?(n) < e(u)) . Then, up to isomorphism, thesimple objects in the heart eX are precisely the objects eV(n) E V(n)[-!(I)], I E A.

PROOF. The first statement in (a) is immediate from properties of D together with the definition of the t-structure given in (3.1). To establish that

Do ‘rCo Z -e z,~ o D , it suffices to show that Doe zCo o D C -e~,O. However, using ihe fact chat the truncation operators can beinterpreted as certain adjoint functors [BBD, (1.3.3i)], elementary properties of the duality D show

e that the functor Do zlo oD: Db(E?) + -e 9” is left adjoint to the inclusion

functor -e.92o ---) Db(E) , so that D o er,o o D g -er,o, as desired. The

assertion that D o ez<o E -e~<o o D follows similarly. The final claim in (a) is an immediate con&quence>f these facts.

To prove (b), observe that HomiaCg)(’ v(n), A(u)) # 0 if and only if


I = u and II = -e(d). Thus, we obtain that ev(J) E Ob(e9’o). Now assume that

HomiaCg)(V(v), ‘k’(L)) = Hom$$;(V(v), P’(n)) # 0.

Then we have n 2 a(n). Also, using the dual of [CPSl, (3.8)], we see Y I I, so it follows that n 2 /(I) 1 a(v) , We have therefore shown that

eV(A) E Ob(‘X) .

To show ‘8 is finite, we use induction on the cardinality of A to show

that each object in eZ’ has finite length. Let A E A be maximal, and put

l7 = A - {A}. Let eX[r] (resp., ‘Z’(n)) denote the heart of Db(‘Z’) (resp.,

&‘(Vect,) 2 Db(J)) relative to the t-structure induced by C on r (resp., {A}) . Following [BBD, (1.4.15)], if T is one of the functors j! , j* , j* , i* ,

i, , i! in(3.1.1),weput eT=CHooToe,where E denotestheappropriatein-

elusion functor ‘X[r] -+ d(iqr]) , eX + d(g), or eX(J) + Db(Vect,) . Using [BBD, ( 1.4.16), ( 1.4.17)], we obtain a recollement diagram

(3.2.1) ex[r] ‘z e x g esqn) 1 i! f t 2

of abelian categories. (By this we mean that the properties [BBD, (1.4.16), (1.4.17)] hold for the functors in the above diagram; see also [P, 511.) Given

A E Ob(eZ’) we have, using [BBD, (1.4.17)(ii)], an exact sequence ej,ej*A --)

A + e ice i* A + 0 in ‘Z’ . By induction on the cardinality of the weight poset,

’ i*A has finite length in eX’[r] . Since the exact functor ‘i* maps simple

objects to simple objects [BBD, (1.4.26)], e ice i* A has finite length in eX .

The category eX’(J.) is semisimple with exactly one simple object k[-e(2)].

To prove that A has finite length, we must show that e@[-f?(l)] has finite

length in e9?‘. However, [BBD, (1.4.26), (1.4.22)] give an exact sequence

(3.2.2) 0 ---* Q + egd-e(41 + i*&W)l + 0

with Q in the image of e i, (and so of finite length). Thus, &k[-c (A)] is

simple, and ‘X is finite. By [BBD, (1.4.26)], the simple objects in eZP’ are the images of the simple

objects in ‘Z’[r] under the full embedding ei, , together with the object j!+k[-!(A)] . Therefore, by induction, to show that the endomorphism algebra

of each simple object in eZ’ is isomorphic to k , it is enough to verify this assertion for j!,k[-l(l)] . Since j,,k[-C(1)] is the image of a natural

morphism ej!k[-l(2)] + ej,k[-f(L)], it clearly suffices to show that the

space of homomorphisms e j!k[-C (A)] + e j,k[-C (A)] is one-dimensional.

SIMULATING PERVERSE SHEAVES

However, using the adjointness properties of (3.2. l), we have

H~rn~(~jJc[-~@)], ‘i&[-e(n)]) z Horn, ,,,,(ei*Li!w(41 9 M-~(~)l) Z End, ,,,,w-w)1) 2 k *

Thus, we have established that the abelian category ‘X satisfies conditions of (1.1).

The remaining verification that ‘Z’ is a highest weight category with the indicated Weyl modules is essentially given in [PS, (5.9)]. In fact, the object T constructed there (by taking successive maximal extensions by Weyl mod-

ules) is easily seen to be a projective generator for ‘Z’. Here we use the fact

that for objects A, B in I%’ we have Extf,(A, B) S HomLb(,,(A, B) . This follows because the heart for any t-structure is stable under extensions [BBD, (1.3.6)]. We leave further details to the reader. (We will not make any use of this fact in this paper.)

To establish (c), assume that ! is compatible in the sense defined. We again give an inductive argument on the cardinality of the weight poset. Clearly, the result is true if IAl = 1 . As observed above, the simple ob-

jects of ‘Z’ are the e i,L , for L simple in ‘X[r] , together with the object

j,&-!?(n)]. By induction, the e V(V) Z ‘V,(V), v E F, are the simple ob-

jects in eR’[F]. On the other hand, the functor i, : d(E[r]) -+ d(lF) is

t-exact, by [BBD, (1.4.16)(a)], and therefore we have ’ i* = i, . We conclude

that the e V(V) Z i*e V,(v) , v E F, are still simple objects in eZ.

We must next show that &k[-e(A)] E ‘V(n). By (b), e V(J) Z j&[-t(1)]

belongs to ‘Z’, so that eV(n) % ‘H’(@[-e(L)]) 2 ej,k[-e(A)]. Thus, if

eV(n) is not simple, we have in (3.2.2) that Q is a ndnzero object in the

image of ’ i, . Hence, there is a nonzero homomorphism ’ V(Y) + e V(n) for some u E F . Necessarily, this means that v < 1, whence 1 (v) < c(n) . This is absurd, since Horn”“” V(v), V(n)) = 0 for n < 0. q

Observe that the length functions C and -e do not play a symmetric role in (3.2(a), (b)). For example, if ! is compatible with the poset A, then -e

is not compatible (unless 4 = 0) and the simple objects of -lZ are the various A(o)[C(o)] , w E A.

In addition, the natural “realization functor” ob(lR’) 3 Db(@7) of

[BBD, $3. l] associated to the heart eZ’ is not generally a full embedding. For example, let G?? be the highest weight category consisting of all finite dimensional (right) modules for the k-algebra of upper triangular 2 x 2 matrices. Then @? has two simple objects a, b , labeled so that a is projective. Define

a compatible length function C by setting !!(a) = 0 and C(b) = 1 . Then eX

is a semisimple category with simple objects eu Z a and Ed Z b[- l] . In fact,

there are no nonzero HornLb(,, 2 Hom$C,) ‘s between these simple objects.

However, Horn2 b D (VJ (a , b[- 11) # 0 . Hence, real is not a full embedding.

80 EDWARDCLINE,BRIANPARSHALL,ANDLEONARDSCOTT

Using the t-structure defined by the length function, we can give the following alternate description of the subcategory gL . The reader should compare this result with [CPS4, (2.4)]. The vanishing of cohomology in odd degrees is one of the notable properties of intersection cohomology sheaves in the standard examples.

(3.3) THEOREM. Let C? be a highest weight category with finite poset A. Assume that C : A + Z is a compatible length function (as defined in (3.2)(c)). Then an object X in Db(%) belongs to gL if and only if e H”‘(X) = 0 for all odd integers m . Similarly, X belongs to gR if and only if -‘H”(X) = 0 for all odd integers m .

If X E Ob(gL), then X has a finite increasing$ltration

0 = F, c . . . c Fi c Fi+, c . . . c Ft = X

(in thesenseof [BBD, $3.11) withsections Grf(X) E Fi/Fi-, E eV(li)[-2ni], for integers ni satisfying n, I a.. < nt and li E A.

PROOF. Since e Ho is a cohomological functor [BBD, (1.3.6)], it is im-

mediate from the definition of gL and (3.2b) that if X E Ob(k?) , then

e Hm (X) = 0 for all odd integers m . Conversely, we suppose that this con-

dition holds for an object X in Db(%7) and we will show that X belongs to

iFL. In this proof, say 2 E Ob(Db(@)) has C-cohomological length equal

to a nonnegative integer m provided that exactly m of the cohomology groups e H’(Z) are nonzero. Since @? has finite global dimension [PS,

(4.3), P, WW41, and any Z E Ob(Db(Q) is represented by a bounded complex, there exists no such that, for each I and each m < f!(n) + no,

Homm( V(n), Z) = 0. Thus, Z E Ob(e&no) and eHi(Z) = 0 for i < no.

Similarly, there exists n, such that e H’(Z) = 0 for all i > n, . It follows every object Z has finite C-cohomological length. Also, by (3. l), the t-structure is nondegenerate, so that [BBD, ( 1.3.7)] implies that Z has C-cohomological length 0 if and only if Z = 0.

Choose n minimal so that Y G ‘r+X # 0. Then Y E er,,er<nX E

Ob(e,%[-n]) and, we have, by [BBD, i1.31, a distinguished tr&gle- Y +

X + e r,,X + . Therefore, ‘H”(X) g Y[n] and eH”(X) S eHm(e~,nX) for all m # n , and n is even. Since C is assumed to be compatible, (3.2~)

implies that Y[n] has a filtration (as an object of eR’) having sections of

the form e V(k). Thus, Y belongs to gL. Finally, er,nX has shorter C-

cohomological length and nonvanishing ’ H-cohomology groups only in even degrees. By induction on the C-cohomological length, er>nX also belongs to

gL. Therefore, by the construction of gL, X belongs to gL. A similar argument establishes the claim concerning gR. Finally, the

assertion concerning the filtration of X follows from the above arguments.


In fact, begin with a filtration of Y with sections of the form e V(n,)[-n] . By

induction on the C-cohomological length, we can assume that X/ Y E ’ r,,X

has a filtration with sections of the form e V(-vj)[-2mj] where n/2 < m, 5 .-.. 0

Using the above results, together with (2.4), we obtain the following consequence.

(3.4) COROLLARY. Let $5’ be a highest weight category with a finite weight poset A and a duality D. Assume that $37 has a Kazhdan-Lusztig theory relative to a compatible length function ! : A + Z.

Let X be any object of E such that eHn(X) vanishes in all odd or all even degrees n , and such that e Hn(DX) E D(-e H-“(X)) vanishes in all odd or all even degrees n . Then X is completely reducible.

We mention here the following corollary of (2.4). We view it as a limited analog of Gabber’s decomposition theorem [BBD, p. 1; Theorem 6.251. (In this analogy, the functor F below should be viewed as the direct image functor for a proper morphism; see also (4.9).)

(3.5) THEOREM. Let ‘8 (resp., 9’) be a highest weight category with weight poset A (resp., A’) and length function C : A + Z (resp., f?’ : A’ + Z) . Also, assume that both G? and g” have a Kazhdan-Lusztig theory. Let F : Db(‘S?) + DB(@) be an exact functor such that

F(c@(‘%?) n Z”(‘Z)) c &(‘@) II ZR(‘%+).

Then, for any weight I E A, the object F(L(L)) is a completely reducible object in g” .

PROOF. Apply (2.4) to the object X = F(L(I)) . We are given the condition (a) of the theorem holds by hypothesis. Since %? has a Kazhdan-Lusztig theory, the hypothesis (b) is automatically satisfied. Therefore, X is completely reducible. 0

(3.6) EXAMPLE. PERVERSE SHEAVES. Let X be a flag variety Gc/B, for a complex semisimple Lie group G, . Then X is equipped with a natural Schubert stratification 9. The category @ of perverse sheaves on X with respect to the middle perversity and with cohomology constant on Schubert strata forms a highest weight category [PS, $51. The odd-degree vanishing of intersection cohomology sheaves, proved by Kazhdan-Lusztig [KL2, (4.2)], shows, using [CPS4, (2.4)], that ‘8’ has a Kazhdan-Lusztig theory. Similar results apply to generalized flag varieties (with finite dimensional cells), but we omit further details.

It is the above example on which (1.2) is modeled. Two further remarks are in order. First, the proof of [KL2, (4.2)], which we used in (3.6) to prove that standard perverse sheaf categories have a Kazhdan-Lusztig theory, already uses much of Deligne’s theory of weights, upon which Gabber’s


decomposition theorem is based. So (3.5) does not yet represent any radical new economy of proof in this context. (Though, conceivably, one might establish the required even-odd vanishing for standard examples combinato- rially using triangulations.) Second, in module category situations, it seems better to go back to (2.4), since the operators which arise as possible F’s are often not known to be everywhere defined. We discuss this further in the next section.

4. Hecke operators

In this section, we define the notion of an abstract Hecke operator on a highest weight category ‘55’ having a length function C : A + Z. The operators in question are very specific, associated with “weak reflections” as defined below, and have the effect of producing character formulas for simple modules indexed by a given weight in terms of those with lower weights. (No other type of Hecke operator will be defined or considered in this paper.) If %? has enough Hecke operators, then % has a Kazhdan-Lusztig theory, and its Kazhdan-Lusztig polynomials can be computed. Even when it is known that 157 has a Kazhdan-Lusztig theory, Hecke operators may provide the most practical method of computing these polynomials.

In practice, it is easy to exhibit candidate Hecke operators, but difficult to show they have the properties really needed. Operators with all the required properties arise in nature geometrically as “push-forward, pull-back” operators on perverse sheaves, cf. (4.9). These operators satisfy the crucial “decomposition property” by virtue of the deep theorem of Gabber mentioned above. Algebraically, operators with the other required properties below can be produced from constructions involving Jantzen translation operators [J, $11.71; see (4.8.3). We show in (4.6) that these operators also satisfy the decomposition property in the presence of a Kazhdan-Lusztig theory in our sense. An abbreviated, nonaxiomatic, version of the argument was given in [CPS4] in treating the main example of algebraic groups.

We begin with the following combinatorial notions. (4.1) DEFINITION. Let A be a poset having a length function ! : A + Z . (a) By a weak reflection on A we mean a pair (A$, s) consisting of a

nonempty ideal As c A and a function s : A$ + A$ such that s2 = id and for any 1 E As either Is 5 1, or 2s > 3, ; in the latter case we assume !!(ns) = L(1) + 1 . (For economy of notation, we will often denote a weak reflection (A,, s) by the associated mapping s .)

(b) Suppose that S is a set of weak reflections on A. For A, v E A, write v ts 1 (or just v 7 L if S is clear) provided there exists a chain u=uo<ul <-.. < u,, = u in A such that for 0 5 i < n there exists si E S with vi E As, and uisi = ui+i .

(c) Let S ‘be a set of weak reflections on A. We say that A is weakly S- connected provided given any 1 E A, there exists a minimal element u E A such that u t A.


The following definition of a pre-Hecke operator is new, but sources of inspiration include Irving’s axiomatization (see [12]) of Jantzen reflection functors, Andersen’s treatment [A21 of Vogan’s work, and Casian’s efforts [Cl, in the context of I&c-Moody Lie algebras, to extend Vogan’s operators Ua (similar to our fiS’s below) and identify them geometrically. Our operators /I, , which keep track of degree information through the translation operator in the derived category, play a role similar to the reflection functors S, in Irving’s work [12], where degree information is tracked through filtration degrees. However, our operators are quite different from those of Irving. For example, his operators have no corresponding geometric interpretation for perverse sheaves (see (4.9)), and they require the validity of the Vogan conjecture (equivalent to the Kazhdan-Lusztig conjecture) before they can even be defined on his filtered categories.

(4.2) DEFINITION. Let % be a highest weight category with duality D : 55 + g and with a length function C : A --) Z . Consider the following data (i)-(ii):

(i) A weak reflection (AS , S) on A . (ii) Let 8 E @(AS) c Ob(@) be the set consisting of all objects isomorphic

to some k’(n), A(1), or L(1) for 1 E AS. We assume that we are given a

mapMu

B, = B : 6 + W&y ,(g’>) , s

where &,Az,( $Y denotes the relative derived category in Db(SZ) associated ) to QA,] .

Then we say that /?, is a pre-Hecke operator (of type s) provided the following conditions (a)-(c) are satisfied:

(a) /3,0(X) c D/?,(X) for any X E 6. (b) Let A E AS . If ;Z < IS, there is a distinguished triangle

v(ns) + &W> + v@>[ll +;

if 1> IS, there is a distinguished triangle

v(n)[-l] + &v(n) + V&s) --);

and finally if As = A then

/3,V(A) 2 v(n)[l] @ v(A)[--11.

(c) Let I E AS satisfy ls > 1. If L(Iz) E Ob(gL), then the adjoint condition

HOm;bp~(B,~(~) > A(y)) z HOm;bcg+(~) 3 P,Ab’))

holds for all weights v E A, and all integers n . Finally, if $ is a pre-Hecke operator, then we say /3, is a Hecke operator

if, in addition, p, satisfies the following decomposition property:

(d) Let Iz E A, be such that I < IS and L(L) E Ob(@‘) . Then /3&(n) is


a direct sum of objects L(V) with v E As . (4.3) REMARKS. (a) Keep the notation of Definition 4.2. For a weak re-

flection s , the full embedding i, : %?[AJ + $9 induces a full embedding

i, : Db(QAs]) + Db(%?) of derived categories. Also, i, defines an equiv-

alence from Db @[AJ) to the full subcategory D&,, ,(5%‘) of Db (%?) . This s

follows from [CPSl, (1.3), (3.9)]. (b) Let /3, be a pre-Hecke operator associated to a weak reflection s .

Let I E A, be given. For the convenience of the reader, we record the various possibilities for /3,&n) . These are obtained by applying the duality functor D to the distinguished triangles in (4.2b) above. If 1 < Is, there is a distinguished triangle A@)[-1] + /?,A(n) + A(Ls) + ; if J > Is, there is a distinguished triangle A(ls) + /?/(A) + A(A)[l] + ; and finally if 2s = A, then /3/(n) % A(A)[ l] @ A(IZ)[- l] .

Our assumption of a duality in (4.2) is largely a matter of convenience to insure these relations (and the dual of (4.2~)). Many of the results of $4

could be recast without this assumption by working with both ZYL and gR . (c) Assume that /I, is a pre-Hecke operator and 1 E A, . Then

Hornbbcg) (/?&(A) , A(v)) = 0 unless Y E A, . To see this, suppose that

v 4 As. We can, replacing A by the ideal generated by As and u , assume

that v is maximal in A. By [CPSl, $31, the quotient functor j* : Db(‘%?) + Db(%‘(v)) has a right adjoint j* : Db(%(v)) + Db(E) . We have A(v) % j,k (as noted in the proof of (3.1)). Thus,

HOm;bcgj(&L(~) 7 A(v)) = HOm&&$L@) > j,k)

2 Hom;b~g~v,,(j*~sW, k) = 0

since J?&(n) belongs to the image D&,, ,(@) of the inclusion functor i, : s

Db(Wsl) + Db@=), as remarked in (a) above.

(d) In part (b), assume also that Is > I and that L(1) belongs to $?’ . Then (1.2), the axioms for a pre-Hecke operator, and (c) above immediately imply that /3&(n) is isomorphic to an object in $?Y for any pre-Hecke op-

erator a,. In addition, it follows [CPS& (2.4)] that if L(L) belongs to &

(resp., &[l]) , then /?,L(n) belongs to &[l] (resp., SL) . Thus, we can rephrase the adjoint condition in terms of (2.0.6) as follows: Let J E As

satisfy As > 1. If L(A) E Ob(@‘) , then

(&wN 9 [A(v = W(~)l~ [P,441) for all weights v E As .

For weights v < 3,) we put

P(V) 1) = dim, Ext&L(rZ) , A(v)).

Thus, by (2.0.3), ,U(Y ,A) is the coefficient of t in the Poincare polynomial

d,i, . We have the following result concerning the explicit decomposition of

&w *


(4.4) PROPOSITION. Consider a highest weight category 5?? as in (4.2). Let s be a weak rejection, and let /3, be a pre-Hecke operator of type s . Let

Iz E A, be a weight such that A < Is, L(2) E Ob(gL), and P,L(A) is completely reducible. Then we have

(In particular, this decomposition holds if 18, is a Hecke operator and A is any

weight such that L(1) E Ob(gL) and A < As E AS .) Finally, L(As)[-[(As)]

lies in gL.

PROOF. Let X = l3,L(A) . By (4.3d), X E Ob(@?‘) , so that for any weight

v , the Poincare polynomial p,” 1 is a polynomial in t . We may compute

the constant term p,” x (0) = d’ ’ imk Homg (X , A(v)) by using the direct sum decomposition of X ‘into simple objects, and we find that this dimension is just the number of summands isomorphic to L(V) .

Since L(A) E Ob(@‘), (4.3d) implies that X E Ob(gL). The Poincare

polynomial p,” x = ([Xl, [A(V)]) may be computed by the adjoint property.

We find for v k As that

(4.4.1) P,L,X =

i

& +ds,n 9 vs>v,

t-‘pfJ + PyL,J 3 VS<V,

tp,“,, + t-lpf 1, us=v.

For v f As, P,” x = 0 by (4.3~). Setting v = Is in the above recursion, it follows that the multiplicity of L&s) in &L(A) is one. Similarly, the multiplicity of L(1) is zero. Now assume v $ {A, As}. If us > v ,

then p,“, JO) = p,“, n(O) = 0 , so L(V) does not appear as a summand of X in this case. Pinally, if vs 5 v , since vs # A, the multiplicity is

dimk Ext,$(L(A), A(v)) = P(V) A). In particular, when this multiplicity is nonzero, )3. > v .

The final assertion follows from (4.3d), so the proposition is proved. q

(4.5) REMARK. Suppose that /3, is a pre-Hecke operator associated to a

weak reflection (As, s) . Then /3, induces a natural Z[t , t-‘l-linear operator

[/3,] on the left Grothendieck group Kt(B[h,] , 4 II\ ) c K~(8, a) . This ac-

tion is defined on the basis vectors [V(n)] by setting [/3,][V(n)] = [/3,V(n)] .

Suppose that 1 E As, 1~ As, and L(A) E Ob(gL) . The relations (4.4.1) for X = Is,L(A) remain valid since they depend only on the adjoint condition (4.2.~). This fact, together with (4.3d), implies that [/9,][L(A)] = [P,L(A)] . Also, observe that although the weak reflection s might conceivably have several pre-Hecke operators attached to it, these pre-Hecke operators all define the same action on the Grothendieck group Ki(S[A,] , f!lAs) .


We also record the following further fact: The operator [a,] satisjies the identity

ra,1’ = [~J11+ [$I[-11 on $(!ZY[h,] , ! II\ ) . This identity can be verified on the Z[t , t-‘]-basis of

K,Lra~J 7 4, > consisting of the [ V@)][-e (A)]] .

It is worth mentioning that in standard cases (such as those discussed in (4.8)) if /3, is a pre-Hecke operator, then [p,][L(v)] = [L(v)[ l]] + [L(v)[- l]]

for any weight v E As for which vs < v and L(V) E Ob(@‘) . Also, the

operators [p,] acting on a suitable union of the groups Kt(O[A] , e) define a module for the Hecke algebra [CPS4, (5.6)].

We now have a result concerning when a pre-Hecke operator is a Hecke operator.

(4.6) THEOREM. Let @Y be a highest weight category with duality D: 529 -+ SF and length function e : A -+ Z. If F has a Kazhdan-Lusztig theory, any pre-Hecke operator is a Hecke operator. More generally, assume, whenever b’ , v E A with f?(e) z l(v) (mod2), that

(4.6.1) E&L(v), A(6)) = E&V(B), L(V)) = 0.

Then any pre-Hecke operator on 5F is a Hecke operator.

PROOF. Assume that (4.6.1) holds for all 8 , v E A satisfying C (0) - C(V) (mod 2) . Let s be a weak reflection and let p, be a pre-Hecke operator of type s . We will apply (2.4) to X = j?,L(A) for 1 E As satisfying Is > 2

and L(1) in G?.

By (4.3d), X belongs to ‘5?, and X[-e(n) - l] E Ob(gL n ZR), since, by (4.2a), DX 2 X . Suppose that u and 0 are two weights such that

Homg( V(y), X) GZ Homg(X, A(v)) # 0

# Horn&X, A(8)) 2 Homg(V(B), X).

This implies that C(V) = e (0) (mod 2) . By (4.6. I), the vanishing conditions required in the hypothesis of (2.4) are satisfied. Therefore, (2.4) implies that X is completely reducible, and & is a Hecke operator.

Finally, if $7 has a Kazhdan-Lusztig theory, (2.1) and [CPS4, (2.4)] imply

E&V(B), L(V)) = 0

if 0 , u E A satisfy C (f3) = ! ( ) u mod 2. Thus, any pre-Hecke operator is a Hecke operator by above. q

Let ! : A + Z be a length function for a highest weight category %Y with a duality. The sets 6,, and BHck of all pre-Hecke and Hecke operators on

@? determine sets Spre and SHck , respectively, of weak reflections.

(4.7) DEFINITION. Let %? be a highest weight category having a duality D and with length function C : A + Z . We say that S!? has enough pre-Hecke operators (resp., enough Hecke operators) provided A is weakly Sr,,- (resp., S,,,-) connected.

SIMULATING PERVERSE SHEAVES a7

Clearly, if @ has enough Hecke operators, then @? has a Kazhdan-Lusztig

theory. (Notice that if 1 is minimal, then L(n)[ -C (A)] belongs to ZYL n ZR .) In 55, we shall prove that under suitable conditions that the presence of enough pre-Hecke operators implies that %? has a Kazhdan-Lusztig theory.

(4.8) EXAMPLE. Let G be a semisimple algebraic group as in (1.1.5). Let C denote the bottom p-alcove, and put Cz = C n X(T) (resp., cz =

c n X(T)) for the set of integral weights lying in C (resp., the closure c) . Clearly, any I E C, is a dominant integral weight. Also, Cz is nonempty if and only if p 2 h , the Coxeter number of G . If s E Z (the simple reflections), let F, be the corresponding face of the alcove C . Then, if F is a face of any alcove C’ = w. C . We say that F is an s-face provided that F=w-F,.

Assume that p > h , and fix A E Cz . We consider the poset (0:) t) associated to 1. There is a natural left action of the (ordinary) Weyl group W on the set WP . il given by z(w . A) = zw ~13. , w E WP , z E W . The

set 0: identifies with the space of orbits WP - n/IV for this left action of IV. By this identification, there is a natural right action of the affine Weyl group on 0:. Thus, for r = w - 1 E X(T)+ and y E WP , we set ry equal to the unique dominant weight in the (left) W-orbit of wy . I. There can be fied points for this action. For example, if s E 1, then rs = r if and only if the s-face of the alcove containing r lies on the hyperplane with equation (x + p , a!“) = 0 for some simple root ff .

Let w E WP . By [Bo, Theorem 1, p. 741, we have I > e(w) for all

s E PVnE if and only if w-l E 0;. Therefore, the set Q of elements w E WP

such that w -1 E 0: is the set of distinguished right coset representatives for the parabolic subgroup W of WP (see [Bo, Chapter 4, Example 3, p. 371). Also, the map w H W-A, w E !2, defines a bijective correspondence between R and 0:. We regard !2 as a poset by restricting the Bruhat ordering on WP to Sz . We can now prove the following result.

(4.8.1) PROPOSITION. Let il E C, be as above. (a)Fory, w~Q,wehavey<w ifandonlyify-Atw-A,i.e.,the

Bruhat ordering on Sz identijes with the t ordering on 0:. (b) Define e : 0: + Z by e (w - A) = e(w) . Then C is a compatible length

function on theposet 0: (i.e., Y < 1 implies e(v) < e(A)). (c) For s E I;, consider the weak reflection (( Ol)S , s) on 0; obtained by

putting (Ol)S = 0: and defining (w . A)s via the right action of WP on 0: defined above. Then, if 9 denotes the corresponding set of weak reflections, the poset 0: is weakly Y-connected in the sense of DeJnition (4. lc).

PROOF. We first prove (a). Suppose that y , w E Q satisfy y - 1 t w .1. We can assume that w = ty for some reflection t = s, np E WP satisfying

a > 0 and (y *1+ p , a”) < np . For a p-alcove C’ , let d(C’) be the number of reflecting hyperplanes separating C’ and C. By [J, $11.6.61, d(y . C) <


d(w . C) . By [Hu, 554.4, 4.51, we have that d(z . C) = e(z) for any z E a. Thus, e(y) < e(w) . Now using the definition of the Bruhat order given in [Dl], we see that y < 20, as desired.

Conversely, suppose that y , w E Sz satisfy y < w . We wish to prove that y. A 1 w -L . By [Dl, (3.8)], we can assume that C (y) = ! (w ) - 1, and therefore that w = ty for some reflection t . Then [Hu, 554.4, 4.51 again implies that d(y . C) < d(w . C) , so that y -2 t w .Iz by [J, $11.6.61, proving (a). It is now obvious that the length function defined in (b) is compatible with the poset structure on 0: .

To prove (c), consider w . ;Z E 0: . For s E E, if !((w -2)s) > C(w .A), then (w .n)s = ws.I2, (i.e., ws E Q) and so by (a), C((w -n)s) = !(w -2) + 1 . Thus, the pair (0: , S) defines a weak reflection on 0: .

To show that 0: is weakly Y-connected, we wish, for w E 0, to connect w . I to the minimal element 1 by a chain as in (4. lb). Choose s E C such that I = a(w) - 1. By [Dl, (3.1)], ws E R and, again by [J, @11.6.6] and [Hu, 554.4, 4.51, ws . I t w . I. It follows, by induction on C(w) , that 0: is weakly Y-connected. q

Let %I be the full subcategory of the category of (finite dimensional) rational G-modules whose objects have composition factors of the form L(7)

for some 7 E 0:. Also, fix u E c, such that u lies on the face F, for

some s E E. We consider the Jantzen translation operators Ti and T[ ,

and put S, = T,” o T[ : FL + gL . Thus, 8, is an exact functor commuting with duality, which is (left and right) adjoint to itself. (For details concerning these operators, see [J, $11.71.) It follows, for any M E Ob(gL), that we have morphisms

(4.8.2)

defined using the adjunction morphisms associated to the adjoint

pairs (T,” , T[) and (T[ , Tf) . Fix 1 E C, , s E E, and v E cz as above. Let A be a finite ideal in 0;

and assume that A, c A is a nonempty ideal stable under the weak reflection

s on 0:) so that (A,, s) defines a weak reflection on the poset A.

(4.8.3) THEOREM. Let A and s G (A,, s) be as in the previous paragraph. For M E 6 z @(As) (see (4.2)) we have E(M) o 6(M) = 0 in (4.8.2). There- fore, we define /I, : 6 + Ob(Db(gL)) by putting, for A4 E 6, /?&II equal to the sequence M + 8,M + M regarded as a complex concentrated in degrees -1, 0, 1. Then p, is a pre-Hecke operator on the highest weight category %?==[A].

PROOF. Fix a weight 7 = w -;1 E AS. First, suppose that 7s s ws - A > 7.

The module S,( V(n)) has a filtration described by the short exact sequence

(4.8.3.1) 0 + V(7s) + cy(7) + V(7) + 0


by [J, $11.7.1.131. In addition, 8, V(r) is checked to have head isomorphic to L(r) . (The only other possibility is that L(rs) is in the head, but this is im- possible since Hom(B,V(r) , L(rs)) = 0 by [J, gsII.7.6, 11.7.151.) It follows that the adjunction map E+, V(r) + V(r) is the natural surjection defined by the above short exact sequence. If the composition V(r) + Q, V(r) + V(r) of adjunction maps is nonzero, then the sequence (4.8.3.1) splits, and S,V(r) has L( rs) in its head, a contradiction. Therefore, /I, V(r) is a complex. Sim- ilarly, /3&(r) and fi,A(r) are also complexes, as required. Clearly, (4.2a) holds with these definitions.

We use the (injective) homomorphism V(rs) + G,V(r) defined in (4.8.3.1) to define a morphism V(rs) -+ /3, V(r) of complexes. Clearly, the associated distinguished triangle is precisely the distinguished triangle V(zs) +

B,W + V(~)[ll + as required by (4.2b). Next, assume that rs < r . It is enough to interchange r and zs in

(483.1) and observe that e,V(rs) % 8,V(r) . Thus, we obtain a distinguished triangle V(r)[-1] ---) /?,V(r) + V(rs) -+ .

Finally, assume that rs = r . Then 8, V(r) = 0 by [J, $11.7.131. Thus, it follows that /I,V(r) s V(r)[-1] @ V(r)[l] .

We now turn to the verification of the adjoint condition in part (c) of the definition of a pre-Hecke operator.

Fix c, u E As , and assume that L(c) E Ob(@‘) and cs > [. We wish to prove that

HOm;b&&%) > A(v)) = HOm;bcg,(%i-) 9 &A(d)

for all integers n . (Observe that following proof establishes the stronger assertion that the above adjoint equality holds without the restriction cs >

c.> First, for arbitrary M, N E Ob($?[As]) , we form the diagram

Hom(M, N) - Hom(B,M, N) - Hom(M, N)

(4.8.3.2)

Horn(M) N) - Hom(M, 8,N) - Hom(M, N)

Here the horizontal maps arise from the adjunction maps defined as in (4.8.2). The center vertical arrow is the adjunction isomorphism which exists because 9, is its own adjoint. By [GJ, (4.1.6)] (and its dual) this diagram is commutative and functorial in M and N .

If we apply (4.8.3.2) to an injective resolution I’ of N and use the functo- riality of the commutative diagram, we obtain another commutative diagram (4.8.3.3)

RHom’(M, N) - RHom’(8,M, N) - RHom’(M, N)

RHom’(M, N) - RHom’(M, 8,N) - RHom’(M, N)


Setting M = L(c) , N = A( ) u in this diagram and taking cohomology yields a commutative diagram valid for all integers n 2 0 : (4.8.3.4)

Hom”W), 44) - Hom”(e,W, 44) - Hom”W), N4)

H4W,W)) - Hom”(W, @,-W)) - Hom”(UL 44) The three terms of the complex L(c) + 0&(c) + L(c) representing

p&(c) give rise to a decreasing filtration with sections L([)[-1] , @&(c), and L(c)[l] in grades -2, - 1 , 0, respectively. This filtration induces a filtration on R Hom’(/?,L( 0 , A(v)) which has sections

in grades 2, 1, 0, respectively. (This result is stated in [BBD, $3.1.3.11, while a detailed proof is supplied in [Ill, sV.1.4.91. The argument is a standard one using Cartan-Eilenberg resolutions.) It follows that there is an E,- spectral sequence which converges to the cohomology Hom’(IS,L(~) , A(v))

of RHom’(P,L(I;), A(v)) in which the columns Ey”, E: ,* and EF” are

and

respectively, while all other columns are identically 0. Observe that E2 g E,

since L(c) E Ob(@‘) . Similarly, the complex A(v) + 8,-4(v) + A(v) has a decreasing filtration

whose sections are A(v)[l], esA(v), and A(v)[-1] in grades 0, 1, 2, respectively. As above, we obtain a filtration of RHom’(L(c) , &A(v)) which yields another spectral sequence converging to Hom’(L([) , /3,&v)) whose first three columns are

and

respectively. As above, the other columns are identically zero and this spectral sequence satisfies E2 FS Em .

Moreover, (4.8.3.4) implies that the E,-terms of the spectral sequences are isomorphic as complexes. Thus, the Ez-terms are also the same, and

dim, Hom”(fi,L(c) , A(v)) = dim, Hom”(L([) , B,A(v)) . 0 (4.8.4) REMARK. The diagram (4.8.3.2) is motivated by a similar construc-

tion given in Casian [C, (8.9)]. The verification of the adjoint condition (c) in


the above proof can be derived, as in [CPS4, $51 (with some effort) from the exact sequences stated in [A2, (2.18)(ii)] (and their proofs). These results, in turn, rely on further results of [GJ]. It is also interesting to observe that, in connection with (4.8.2), the rational G-modules M for which the sequence (4.8.2) is a complex clearly form an abelian subcategory of the category of rational G-modules. (4.9) EXAMPLE. GEOMETRIC HECKE OPERATORS. Perhaps the canonical

model of a Hecke operator is the functor a*Ra,[l] on perverse sheaves, where n: G/B + G/P is the fibration obtained from a minimal parabolic subgroup P of a complex semisimple Lie group G with Bore1 subgroup B c P . The critical decomposition property is a consequence of Gabber’s decomposition theorem [BBD, p. I]. It follows also from (2.4), together with the “even-odd” vanishing proved in [KL2, (4.2)], though the latter uses already much of the machinery of the theory of the Weil conjectures.

The relationship of the geometric operator with the algebraic one treated above is discussed by Casian [C] in the Kac-Moody context, where it is as- serted that the constructions correspond under favorable circumstances.

(4.10) EXAMPLE. QUANTUM GROUPS. In the setting of ( 1.1.6), the discussion of (4.8) for U, carries through in this case using the results of [APW] on translation functors. Similarly, the results hold for SL,(n) (and arbitrary characteristic), since the necessary results on translation functors can be easily established using [PW, Chapter lo].

(4.11) REMARK. “CANDIDATE" PRE-HECKE OPERATORS. In (4.8), /3,L(,l), fi,A(n) , and /3, V(n) can be characterized independently of their description (see (4.8.2)) in terms of Jantzen reflection functors. This characterization is achieved by describing 8,L(i) , etc. intrinsically. We briefly indicate this description.

Fix r E 0: and a weak reflection s such that r < zs . Let I denote the

ideal in 0: generated by rs , and consider the coideal n = {r , zs} of F . Using results described in (1.1.3), the quotient category %?(a) has exactly two simple objects, namely, a = j*L(7) and b = j*L(zs) . Using further quotient category arguments and results of [J, 511.71, one finds that the projective cover Q of a in ‘Z?‘(n) is a self-dual object having composition factors a, b , a. Recall from [BBD, (1.4.6. l)] that there is a natural morphism f: j!Q --) j*Q . Then it can be shown that 8,L(7) identifies with the image

j!* Q of f . An equivalent approach is to form the extension V(7)’ of V(7)

by V(7s) . (One can show that Ext’ (V(7), V(7s)) is l-dimensional here.) There is a dual extension A(7)’ . Then 8, L(7) is the image of a natural

map V(r)’ + A(rs)‘. One also has CI,V(z) g j!Q Z V(7s)‘, and 8+(z)

is described dually. As usual fi, L(7) , /?, V( 7) , and $A(7) are defined by associated complexes (and the /3,L(7s) , etc. are defined by translating the appropriate complexes).

An advantage of this description is that it makes sense in any highest


weight category with duality, assuming that dim, Extb( V(r), V(rs)) = 1 . Consequently, there will be “candidate” pre-Hecke operators for such categories. (To be pre-Hecke operators, the operators must satisfy the adjoint condition.) It would be interesting to know of examples generated in this way from general classes of finite dimensional algebras (perhaps relaxing the duality requirement in Definition 4.2). The issue, in general, is: What finite dimensional algebras are associated to highest weight categories having valid Kazhdan-Lusztig conjectures as described in $5 below?

5. Abstract Kazhdan-Lusztig conjectures

Let %?? be a highest weight category with finite weight poset A. Let e : A + Z be a length function, and let S be a fixed set of weak reflections (4.1). When 9 has enough Hecke operators (4.7) and S - Suck, the Kazhdan-Lusztig polynomials (2.0.5) are determined recursively by A, !? , and S . (This fact follows from the recursive formulas (4.4. I).) In general, as we demonstrate below, the combinatorial information contained in A, C , and S recursively determines (perhaps overdetermines) an explicit set of polynomials.

In the tradition of the work of Deodhar [D2] and Casian-Collingwood [CC] (see also [L2], [LV], [Ill), we associate to the combinatorial triple (A, ! , S)

a Hecke module J = J(A) ! , S) over the ring Z[t , t-l]. By definition,

M is the free Z[t , t-’ ]-module with basis T, , 1 E A. (We intuitively think of TA as corresponding to [V(n)[-f?(d)]] in the left Grothendieck group

Kt (B , e) . Similarly, the basis element b, below corresponds to the “phan- tom” element [L(n)] in the left Grothendieck group). For each s E S , there is an operator /3: on 1 with action mimicking that of (4.2c), i.e., we define for 1 E As :

i

t-‘(T,+T,) ifIs>1,

(5.0.1) D;Tl = t(Tl+ T,) if13.>Is,

(t + t-‘)Tn ifLs=J.

If I$ As,wemerelyset &T,=O. Assume that the poset A is weakly S-connected. We can attempt to define

recursively elements { bv}vE,, in J by the following formula (inspired by

(4.4)):

i

t-e(A) T if 1 is minimal,

(5.0.2) by = /3;bh _“’ c p’(v , h)b, if3,EA,andL>Is. vs~v<Ls

In this expression, we define $(v , As) to be the coefficient of t in the can-

didate PoincarP polynomial p: ,t E Z[t , t-l] defined recursively by

(5.0.3) b,, = c t-e(“)$v, h TV .


Conceivably, because of the possible existence of different paths to a weight, these formulas may not be consistent. Also, in all examples considered in this paper, we will have p: , d = 0 unless Y I Q . This implies immediately that

the b,‘s, when they do exist, form a basis for .N . This basis property holds

generally, as follows by arguing on lengths: p: d = 0 unless C(V) < C (0) orv=o. (0 ne simple hypothesis on the weight poset A which always guarantees that p: ,D # 0 + v 5 cr is the following: for any weak reflection

s and weights 1, v E A, such that 1s < A and u < 1, then us 5 1. This condition always holds in the examples we consider, but we prefer not to assume it.)

We can also consider candidate Kazhdan-Lusztig polynomials defined in terms of the candidate Poincare polynomials by the formulas

(5.0.4) P’ v,1= tewwpJ

v,l’

Thus, P’(Y , A) is the coefficient of te(n)-e(v)-l in P: I . When the polynomials (5.0.4) exist, they satisfy the recursive relation: ’

Pi,,,+t2Pi,1-(*) ifvs<u,

(5.0.5) P’ v,t =

i ’

Pi 1 + t2PLs,d - (*) if us > u,

(1 + t2)p:,, - (*) if us = u .

for SES, LEA, with A<As,where

(x) = c p’(z, 2)tL(i)-L(TJ+1P;,,. TS<7<l,V

Of course, the recursion starts at 1 minimal with P,’ 1 = 1 . We can read-

ily verify inductively that (when they exist) each candidate Kazhdan-Lusztig polynomial Pi 1 is a polynomial in t2 having t-degree at most e(n) -e (u) - 1 .

Also, P,’ A = 1’ for all weights 1. This discussion suggests the following definition. (5.1) DEFINITION. Let ‘S? be a highest weight category with finite poset

A. Assume that E has a duality and a length function t : A -+ Z . Suppose S is a nonempty set of weak reflections such that A is weakly S-connected. If the construction of the basis {b,} above is possible, then we say that the highest weight category ST has a Kazhdan-Lusztig conjecture relative to (A, e , S) . The corresponding Kazhdan-Lusztig conjecture states that the candidate Kazhdan-Lusztig polynomials coincide with the Kazhdan-Lusztig polynomials (2.0.5):

P’ v,l ‘C,A for each pair of weights u , 2 E A.

Observe that if a highest weight category ‘S’ has a Kazhdan-Lusztig conjecture, then a necessary condition that it be true is that the coefficients of P’ v ,1 be nonnegative integers. Also, we see that if E’ has a Kazhdan-Lusztig


conjecture which is true, then necessarily S? has a Kazhdan-Lusztig theory as in (2.1). This follows from the comment above that each Pi I is a polyno-

mial in t2, together with [CPS4, (2.4)] (discussed after (2.1)). Of course, in the case when the Kazhdan-Lusztig conjecture is true, if Pi, I # 0 , then necessarily u I A.

Regard Z as a Z[t , t-‘l-module through the homomorphism Z[t , t-l] + Z (t H -1). There is a natural isomorphism Kf(E?, e) B~[~,~--I] Z E K,(g)

defined by base change. Put x= J @zll t-~l Z . There is a formal isomor-

phism

(5.1.1) t:JzK;(%J) CT, I-+ [v@)l[-c@)l)

of Z[t , t-‘l-modules inducing an isomorphism z 8 Z: 27 K,(g) . These isomorphisms define a commutative diagram:

A A K;(E’,e)

Let i denote either composite map from .M to K,,(g) . Thus, i( 7”) =

(- l)e’qV(q] . We have the following result.

(5.2) LEMMA. Assume that E? has a Kazhdan-Lusztig conjecture, so that the candidate Kazhdan-Lusztig polynomials P: 1 as well as the basis {b,} of the Hecke module M are consistently defined. ‘Then:

(a) For a weight 1, the conditions L(n)[-e(n)] E Ob(&) and z(b,) =

[L(n)] in K,“(%?, a) hold ifand only if PL,l = PV,n for all weights v . (b) For a weight )3., the condition f(b,) = [L(J)] holds in K,(g) ifand only

if PL n(-l) = P, n(-l) (polynomials evaluated at t = - 1) for all weights v .

PROOF. We first prove (a). Suppose that L(n)[-t!(n)] E Ob(G@) and z(b,) = [L(J)] . Then

[W)l = @,I = c t -P(i)&,nw~) 9 [-C(v)11 = ~PJ,,nvwI, ” ”

holds in K,f($?, e), so that P:,~ = P”,~. Thus, Pi,, = PV,, by (2.0.5) and (5.0.4), as desired.

Conversely, suppose PL , I = P, , I As noted earlier, Pi, I is a polynomial

in t2. Hence, by [CPS4, (2.4)], L(n)[-a(n)] E Ob(J?) . It is now clear that z(b,) = [L(n)]. This completes the proof of (a).

The proof of (b) is similar, using the character formula (2.0.8). 0 Recall that Spre is the set of weak reflections corresponding to the pre-

Hecke operators; see (5.7). We now have the following result.


(5.3) THEOREM. Let %? be a highest weight category with Jinite weight poset A. Assume that S? has a duality D and a length function e : A + Z. Assume that S c Spre is such that A is weakly S-connected (so that @? has enough pre-Hecke operators). Finally, suppose that %Y has a Kazhdan-Lusztig conjecture relative to (A, C , S) . Then the following statements are equivalent:

(a) Forallweights 1, v, wehave Pi,I(-l)=PV,,(-l).

(b) Any pre-Hecke operator p, of type s E S is a Hecke operator, i.e., /3,L(J) is completely reducible for any weights 3, E As with 1 < Is. (Recall that, by (4.3d), /3,L(n) is isomorphic to an object in ‘%’ .)

(c) The Kazhdan-Lusztig conjecture is true for @, i.e., P,,, = PL ,1 Vv , LEA.

PROOF. First, by (4.9, any pre-Hecke operator p, defines an operator [a,]

on the left Grothendieck group K,“(a[AJ , C I* ) by setting [p,][V(v)[k]] =

[/?,V(v)[k]], v E As. Thus, by (4.2b), we haves (for v E A,) :

[Bslu+)l =

i

t-‘[V(v)] + [V(vs)] if vs > v,

W(v)1 + V(vs)l ifvscv,

(t + t-‘ww1 if vs = v .

Using this formula, we verify that

(5.3.1) z O I( = [P,l O z

on the Z[t , t-‘I-submodule J$ of J generated by the TV for v E As. We now show that (b) + (c) by establishing the following somewhat

stronger result (which will be used again later in the proof).

SUBLEMMA. Assume the hypotheses of (5.3), and let r c A be a nonempty ideal. Assume, for any weak reflection s , that p, L( o) is completely reducible whenever sES, BEAM, o < ws E r, and p, is a pre-Hecke operator of type s . Then P, I = PL ,1 for all 1 E r and all weights v .

PROOF. Fix a weight 1 E I. It is clear that P, I = PL L if 2 is minimal. If 1 is not minimal, we can find a weak reflection i and a’ weight w E A, such that w < ws = I, since A is weakly S-connected. Necessarily o E I. By an evident induction on ]I] , we can assume that P, e = PL 0 for all 8 < L and

all v . In particular, it follows that we have an eiuality ‘,u(v , o) = ~‘(v , w)

for all v . Also, using (5.2a), we have that L(@[-C(e)] E Ob(&) and z(be) = [L(e)] for any weight 8 < 2.

By definition (5.0.2), we have

(5.3.2) B,‘b, = b, + c P’(V > 4b,. VSgKO

Let /3, be a pre-Hecke operator of type s , and apply z to (5.3.2). Since


[/~,][L(o)] = [/3s(L(o)] by (4.9, we obtain from (5.3.1) that

The last equality follows from (4.4) (since OS = A) which also implies that

L(A)[-l(A)] belongs to gL . The desired conclusion that P, A = P: A for all v now follows from (5.2a) again. This completes the proof of the sublemma. 0

It is obvious that (c) + (a). It remains to prove that (a) + (b). We suppose that the hypothesis (a) holds, but assume that (b) is false. Let y be a weight minimal with respect to the following property: for some weak reflection s we have ys < y E A, , and, for some pre-Hecke operator /3, of type s , we have that /3sL(ys) is not completely reducible. We apply the sublemma to

the ideal I = (-cc, 7) to conclude that, for o = ys , we have Pi w = P, o ,

’ ’ and, in particular, ,u’(v , o) = P(Y) o) for all weights v .

By (5.2a), L(o)[-t(w)] E Ob(&). Apply z to the expression (5.3.2), and use (5.2a) and (5.3.1), to conclude that

= @,) + c P(V > w)[Uv)l us~v<w

in Kt(B, a). Observe that i(b,) = [L(y)] in K,(F) by the hypothesis (a)

and (5.2b). Applying the homomorphism Kt(‘i%‘, e) + K,(g) to (5.3.3), we find that in K&F’)

(5.3.4) vQw41= [L(Y)1 + c P(V) w)[L(v)l* vs+Kw

On the one hand, for v < o, the multiplicities [j?,L(o): L(Y)] are the coefficients P(V) w) in (5.3.4). On the other hand,

P(V) w) = dimExt’(L(w), A(v)),

by the definition given in $4. Since as is a pre-Hecke operator, we have

(5.3.5) Hom(P,W), 44) g HorW44T &W)

for v < o (and hence v E As) . Suppose first that vs < v andp(v, o) # 0 in (5.3.4). Since L(o)[-C(w)]

lies in ZL, we have e(o) $ C(Y) (mod2) and Ext’(L(o), A(vs)) = 0 all


by [CPS4, (2.4)]. Hence, the long exact sequence of cohomology induced by the distinguished triangle (see (4.3b))

yields

Hom(L(o), Bs4v)> 2 Ext’(L(o), A(v)),

since Hom(L(o) , A(vs)) = 0 because vs # o. Thus, by (5.3.5), we obtain, writing X = p,L(o) , that

(53.6) [X : L(V)] = dimk Horn(X) A(v)).

Secondly, if vs = v and P(Y) o) # 0, then

so Hom(Uo) , BsA(v>> g Ext’(L(o), A(v)),

and the equality (5.3.6) holds for this weight v also. Thus, the multiplicities of the composition factors L(V) of X = B,L(w)

are given by (5.3.6). In addition, X is self-dual by (4.2)(a). We assert that if X E Oh(E) is self-dual and satisfies (5.3.6), thenX is completely reducible. (This will provide a contradiction to our original assumption.) We let v E A be minimal such that L(V) is a composition factor of X . Clearly, Horn(X) A(v)) E+ Hom(X/ rad(X) , A(v)), where rad(X) denotes the radical of X . Thus, L(V) only appears in X as a composition factor of the head of X . However, since X is self-dual, this clearly means we can write X = Xi @ X, , where Xi is a direct sum of copies of L(V) , while L(V) does not appear as a composition factor of X, . By the Krull-Schmidt property for %7, we conclude that X, is self-dual. Since the property (5.3.6) evidently also holds for X, , we have, by induction, that X is completely

reducible. 0 We next establish the following preliminary result.

(5.4) LEMMA. Let %Y be a highest weight category with length function C : A + Z and duality D. Suppose that s is a weak rejlection on A and that /?, is a pre-Hecke operator on @? of type s . Assume L(l.)[-e(n)] E Ob(EL) , 3, E A,, and Is > 1. Then L(ls) has multiplicity one as a composition factor

of&w. PROOF. Let A be the ideal generated in A generated by the weights

6 such that L(6) is a composition factor of jl,L(n) . Thus, /l,L(n) E Ob(E’[A]) . Let w E A be a maximal element, so that, because A(o) is injective in g[A] , we have Hom(/3,L(A), A(o)) # 0. By (4.2)(ii), that A c A,. Using the adjoint condition (4.2)(c) for a pre-Hecke operator, we obtain that Hom(L(A) , /ISA(o)) # 0. The long exact sequence of cohomology, applied to the possibilities for the distinguished triangles involving j?,A(w) listed in (4.3b), implies immediately that w 3 Is . (Here we also use the fact, proved


in [CPSl, (3.8)(b)], that if Hom’(L(<) , A([)) # 0, for weights < # [, then < > c .) Thus, A is contained in the ideal I = {y ] y 3 As} . Hence, A(ls) is an injective object in %?[I]. Therefore, the adjoint condition (4.2)(c) and (4.3)(b) yield, arguing as before, that the multiplicity of L(ls) as a composition factor of B&(n) equals

dimk Hom(/3,L(Z), A(ls)) = dim, Hom(L(1), b&s))

= dimk Hom(L(n), A(A)) = 1.

This completes the proof of the lemma. •I We are now ready to establish the following result, which gives various

conditions equivalent to having a valid Kazhdan-Lusztig conjecture.

(5.5) THEOREM. Let g be a highest weight category with$nite weight poset A. Assume that % has a duality D and a length function e: A + Z. Let

s = spre be such that A is weakly S-connected. For each s E S, Jix a pre-Hecke operator /3, of type s . The following are equivalent:

(a) For each s E S, the pre-Hecke operator /?, is a Hecke operator. (b) For each s E S and weight z E A3 such that zs > z, the simple module

L(zs) is a direct summand of p,L(z) . (c) For each s E S and weight z E As such that zs > z, we have

(5.5.1) Hom(P,W , UN) # 0.

(d) 9 has a Kazhdan-Lusztig theory. (Equivalently, by [CPS4, (2.4)] and the presence of a duality, Ext”(L(v) , A(z)) # 0 implies n = C(V) - e(z) (mod 2) for all weights u , z .)

(e) Whenever 0, u E A satisJL ! (0) - C(V) (mod 2) , we have

(5.5.2) Ext’(V(B), L(V)) = 0.

(f) %? has a Kazhdan-Lusztig conjecture relative to (A, ! , S) and that conjecture is true.

PROOF. First, we assume that (b) holds and show that (d) is true. Because A is weakly S-connected, it suffices, by induction, to prove that, if r E A,

for some weak reflection s such that rs > r and L(z)[-.! (z)] E Ob(@) ,

then L(zs)[-e(zs)] E Ob(&) . (Because % has a duality D, the identity

D& = gR and the assumption that L(zs)[-e(zs)] E Ob(&) imply that

L(zs)[!(zs)] and hence L(zs)[-e(zs)] lie in gR .) By (4.3d), p,L(z)[-e(zs)]

belongs to c@. Hence, the hypothesis (b) insures that the direct summand

L(zs)[-C (rs)] also belongs to ZR . Thus, (d) is true. A similar induction shows that (a) + (b). Obviously, (e) is a special case of (d). Also, (e) + (a) by (5.4). In addition,

(5.3) shows that (f) + (a). Now assume that (a) holds. Thus, % has a Kazhdan-Lusztig theory, since

it has already been shown that (d) follows from (a). We identify the Hecke

1 .


module M = &?(A, C , S) with the left Grothendieck group Kt(8, !) using the isomorphism (51.1). Thus, for each weak reflection s E S the operator

[/3,] on A 2 K:(B) e) identifies with the operator & in (50.1). By (5.4), the elements [L(v)], v E A, in A satisfy the recursive relations (50.2) for the elements b, , u E A. Hence, the basis {b,} for A can be consistently defined, so that %? has a Kazhdan-Lusztig conjecture. Finally, the conjecture is true by assumption (a) and (5.3). Hence, (a) + (f).

Clearly, (b) + (c), and it only remains to show that (c) + (b). Thus, assume that (5.5.1) holds for all weak reflections s E S and weights r E A, such that rs > r . It then follows for any such s and r that L(rs) appears in the head of B&(r) . However, since /I&(r) is self-dual, L(rs) also appears in the socle of fi,L(r) . Using (5.4), we conclude that L(rs) is a direct summand of p&(r). Thus, (b) holds as desired. EI

(5.6) Discussion. The Lusztig conjecture. Let G be a semisimple, simply connected algebraic group as in (4.8). Assume that the field k has characteristic p 2 h , the Coxeter number of G . Fix L E Cz , and consider the poset

(0:) 7) . Consider the set 9 of weak reflections defined on 0: in (4.8. lc). Also, for y , w E Wp , let PY w denote the classical Kazhdan-Lusztig polynomial for the Coxeter group ’ Wp [KLl].

Now let r(n) c 0: be the subset of weights of 0: which satisfy the

Jantzen condition. Observe that I(n) is an ideal in 0: . We will work with the highest weight category q[r(n)] .

For s E 9, we let r(n), be the set of weights r E I’(1) with the property that there exists a weight c E r(n) such that r 5 c and cs < [. Using [Hu, p. 1511, together with (4.8.1), we see that r(n), is an ideal in r(n) which is stable under the operator s . Thus, if r(n), is not empty, we obtain a weak reflection (I’(n), , s) . In what follows, we let 9 continue to denote the corresponding set of weak reflections on r(n) . Clearly, r(1) is weakly 9’- connected, by an application of (5.8.1~). Also, (5.8.3) implies that 9 c Spre, the set of weak reflections associated to the pre-Hecke operators on the highest weight category sl;[r(n)l .

As stated in (2.5), the classical Lusztig conjecture [Ll, 53, Problem IV] describes a character formula for irreducible rational G-modules L(y) with high weight y E r(n) .

We now prove the following result which relates the above Lusztig conjecture with the results in this section.

(5.7) PROPOSITION. Let G be a semisimple algebraic group as in (5.6). Fix 1 E C,. Then the highest weight category %AII’(n)] has a Kazhdan-Lusztig conjecture with respect to (r(n), C , 9). Also, the Lusztig conjecture (2.5) is valid if and only if the Kazhdan-Lusztig conjecture holds for the highest weight category gA;[r(n)l.


PROOF. To check this, we first recast the formula (2.5) in a slightly different form. Consider the opposition involution r H r* E -wo(7) on X(T).

Also, for z E WP , let z* = ~0’ zwo . Then we have, (z - A)* = z* . L* , so that “twisting” the representations of G through the graph automorphism G + G associated to the opposition involution, we can rewrite (2.5) as

chL(w* . A*) = Cy(-l)LCV)--L(w)Pyu’o,u,le (-1)ch V(y* . A*). Since z*wo =

woz , we can therefore, changing notation, rewrite (2.5) once more as follows:

(5.7.1) chL(w-A) =~(-l)~‘y’-“w’p,,,,,o,(-l)chV(y.I). Y

Again, we sum over all elements y E WP such that way 5 wow (equivalently,

y 2 w [Dl, (3.5)]) and y . I E X(T)+. (In particular, using the notation of (4.Q we have y E R , the set of distinguished right coset representatives of W in WP .)

Next, observe that the recursive relations (5.0.5) above coincide with the ones given in [D2, (3.9)]. (The reader is cautioned that we are only using the recursive relations of [D2] , and not the Hecke module discussed there which differs somewhat from our Hecke module.) In applying Deodhar’s results, we take the affine Weyl group WP as the Coxeter group having parabolic subgroup given by the ordinary Weyl group W . Thus, by [D2, (3.4)],

(5.7.2) P’ y-A,w-l = pw y wow * (One has to convert notation from Deodohar’s left-hand action to our right- hand action, but this may be achieved through the identity Py , w = Py- I , w- I for Kazhdan-Lusztig polynomials.) This establishes that %l[r(l)] has a Kazhdan-Lusztig conjecture, and this conjecture is equivalent to the Lusztig conjecture (2.5), as claimed. •I

(5.8) REMARK.~OMPARISONWITH OTHER NOTATIONS. Andersen[A2]and Lusztig [L2] use yet another notation system. Their left-hand “dot” action of the affine Weyl group WP is not the usual “dot” action of [J] we have

used, but instead y . A is defined, for y = si . . . sk , to be the image of 1 under successive reflections through walls labeled sk, . . . , si . Their labeling process is not specified precisely, beyond the observation that the walls fall into orbits, the latter in bijection with the simple reflections.

We now obtain the following result, also stated above in (2.6). It gives an equivalent formulation of the Lusztig conjecture. (Observe that the weights

7, 7s) in the statement of the theorem below are mirror images of one another in adjacent p-alcoves lying in r(n) .) Other equivalences with the Lusztig conjecture are indicated in (5.10) (d), (e), (f).

(5.9) THEOREM. Let G be a semisimple, simply connected algebraic group defined over an algebraically closedJield k of characteristic p > 0. Assume that p 2 h, the Coxeter number of G. Fix 1 E C,. Let l?(n) be the ideal (in the poset (0:) t)) of d ominant weights y = w . ;Z, w E Wp , satisfying


the Jantzen condition. Let 9 be the set of weak reflections defined in (5.6) The Lusztig conjecture (2.5) holds for G if and only iJ whenever s E 9, z < 7s E r(A), , we have

(5.9.1) Ext;(L(r) , L(7s)) # 0.

PROOF. Consider the highest weight category gl[[r(n)]. We use the length function e = e lrcg : r(A) + Z defined in (4.8.1). As observed in (5.6), the set

9 of weak reflections on r(n) is contained in the set SPre associated to the

pre-Hecke operators on %‘&I@)]) . Also, by (5.7), we know that gA[r(n)] has a Kazhdan-Lusztig conjecture (relative to (r(n) , 9, e)) which is equivalent to the Lusztig conjecture (2.5).

Fix s E 9 and 7 E r(n), satisfying 7 < 7s. Write 7 = w - 1. By (5.7) and (5.5), it suffices to show that condition (5.9.1) is equivalent to condition (5.5.1) for %[I@)]. We consider (4.8.2) with M = L(7). Observe that the s-face of the alcove w . C lies in the upper closure of w . C (in the sense of [J, $11.6.21). Thus, by [J, §§II.7.15, 11.7.191, we obtain from (4.8.2) a complex 0 + L(7) --) 8,(L(7)) --) L(7) --) 0 which is exact at either end. We decompose this complex into the short exact sequences

(5.9.2)

and

0 --+ L(7) + 8,L(7) + Q + 0

(5.9.3) O+/I,L(z)-+Q-L(z)+O.

Since B,L(zs) = 0 by [J, $11.7.151, we have that

Hom”(8,L(r), L(7s)) 2 Hom”(L(r), B,L(zs)) = 0 Vn.

Thus, the long exact sequence of cohomology applied to (5.9.2) yields Hom”(Q, L(7s)) = 0 for n = 0,l. Similarly, (5.9.3) finally gives Hom@,L(r),

L(7s)) g Ext’(L(r), L(7s)). Therefore, (5.9.1) and (5.5.1) are equivalent, as desired. q

(5.10) REMARKS. (a) The elementary argument given in the above proof of the equivalence of (5.5.1) and (5.9.1) is essentially contained in [A2]. A second, more conceptual, proof can be based on the arguments given for (4.8.3). Namely, by the spectral sequence arguments there, we have Hom(B,L(r), L(7s)) g Hom(L(r), B,L(rs)) . We see also that pS(7s) S

L(7s)[l] @ L(zs)[-11, since B,L(zs) = 0. Thus, (5.5.1) and (5.9.1) are equivalent.

(b) Assuming the hypotheses of (5.9), it can be proved that

dim, Extk(L(rs), L(7)) = dimk Extk(L(z), L(7s)) 5 1

for any weight 7 E O,+ . To see this, we can suppose that 7 < 7s. An elementary long exact sequence of cohomology argument shows that there

is an injection Exth(L(rs) , L(7)) --) Extk(L(ss), A(r)) . Thus, the desired

conclusion follows since the group Exti(L(rs) , A (7)) is 1 -dimensional, as


proved in [J, §11.7.21b]. Similar remarks apply to the category 6’ as well as to the category gi for a quantum enveloping algebra as in (1.1.6).

Furthermore, it is possible that Ex&L(r) , L(7s)) = 0 once 7 < 7s

lie outside the Jantzen region F(n) . For example, let G = SL, , let 7 =

(p2 - 2)~ , and 1 e s be the reflection associated to the face (p2 - 1)~. t (c) The Kazhdan-Lusztig conjecture [KM] for Verma modules for the

principle block 8,,, of the category B was in the form of a character formula. Work of Vogan [Vl], showed it to be equivalent to an identification of Kazhdan-Lusztig polynomials with polynomials associated to certain Ext- groups, as in (2.02a). In addition, he proved the “even-odd” vanishing (2.02b) of these Ext-groups (and apparently left open in [V2, Problem 7, p. 7391 the question as to whether this even-odd vanishing property itself implied the Kazhdan-Lusztig conjecture). Thus, the highest weight category 8,,” (with the length function e defined by ! (w .( -2~)) = e (w )) has a Kazhdan-Lusztig conjecture, and this conjecture coincides with the original Kazhdan-Lusztig conjecture.

Also, Vogan proved [Vl] the formal equivalence of the Kazhdan-Lusztig conjecture to the complete reducibility of the /3,L(7) for 7 E A, with 7 < 7s.

The assertion that the /3,L(7) are completely reducible is now often called the Vugun conjecture. As recorded in [GJ, p. 2851, Vogan also established that the “even-odd” vanishing property (2.0.2b) is implied by the weaker condition that L( 7s) be a direct summand of /?,L(z) (for 7 E A$ with 3, < A,) . Andersen [A21 proved most of Vogan’s results above for the corresponding case of rational representations of semisimple algebraic groups in characteristic p > 0.

As (5.5) above shows, the “even-odd” vanishing condition does imply the validity of the Kazhdan-Lusztig conjecture in our general setting. At the same time, we have shown that the equivalent assertions are also equivalent to the assertion that L(7s) is a direct summand of j?&(7) for 7 < 7s .

(d) Let G be as in (5.9). From the universal mapping property of Weyl modules, the Lusztig conjecture is equivalent to the following statement [CPS4, 5.41:

Whenever s E 9, 7 -C 7s E r(A), , there is a quotient of the WeyZ module V(7s) having only two composition factors L(7) and L(7s) .

(In the context of 6’) Gabber and Joseph [GJ] have noted that the analog of (5.9.1) is equivalent to a similar assertion concerning the Verma module

V(7s) -1

Using [J, §11.7.18], we see that V(7s) has a unique quotient X(7s) with simple head L(7s) , simple socle L(7) , and such that any other composition factor L(Y) satisfies 7 # u # 7s. Finally, [CPSS] shows that the above version of the Lusztig conjecture can be further reduced to a question involving the group scheme G, T .

(e) Continue to let G be as in (5.9). We have [CPS4, 5.41:


Let 8, u E r(A) satisJL l(O) E C(v) (mod2). Then Ext’(V(8), L(v)) = 0.

(f) We mention that, using the results described in (4.10), the results of (5.7) and (5.9) carry through for the case of quantum enveloping algebras U, of (1.1.6) or to the quantum group SL,( n) with the important modification

that the ideal r(A) can be replaced by any finite ideal of weights in 0: = W, . A. We leave the straightforward verification to the interested reader. Similar remarks also apply to (d) and (e) above in the quantum case, and to at least the first reduction in (f).

REFERENCES

[Al] H. Andersen, The strong linkage principle, J. Reine Angew. Math. 315 (1980), 53-59.

L421 - , An inversion formula for the Kazhdan-Lusztig polynomials for afine Weyl groups, Adv. Math. 60 (1986), 125-153.

i-431 -9 Filtrations of cohomology modules for Chevalley groups, Ann. Sci. Ecole Norm. Supp. (4) 16 (1983), 495-528.

[APW] H. Andersen, P. Polo, and K. Wen, Representations of quantum groups, Invent. Math. 104 (1990), l-59.

[BB] A. Beilinson and J. Bernstein, Localisation des X7-modules, C. R. Acad. Sci. Paris 292 (1981), 15-18.

[BBD] A. Bielinson, J. Bernstein, and P. Deligne, Analyse et topologie sur les espaces sing&ares, Asterisque 100 (1982), 1-171.

[BG] A. Bielinson and V. Ginzburg, Mixed categories, Ext-duality, and representations (results and conjectures), preprint, 1986.

[BCG] J. Bernstein, I. Gelfand, and S. Gelfand, A category of .Y-modules, Functional Anal. Appl. 10 (1976), 87-92.

[Bo] N. Bourbaki, Groupes et algebres de Lie, Chaps. 4-6, Hermann, Paris, 1968. [BK] J. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, In-

vent. Math. 64 (1981), 387-410.

[Cl L. Casian, Proof of the Kazhdan-Lusztig conjecture for Kac-Moody algebras (symmetris- able case), preprint.

[CC] L. Casian and D. Collingwood, The Kazhdan-Lusztig conjecture for generalized Verma modules, Math. Z. 195 (1987), 581-600.

[Cl] E. Cline, Simulating algebraic geometry with algebra. III: The Lusztig conjecture as a TG,-problem, Proc. Sympos. Pure Math., vol. 47, Amer. Math. Sot., Providence, RI, 1987, pp. 504-521.

[CPISI] E. Cline, B. Parshall, and L. Scott, Finite dimensional algebras and highest weight categories, J. Riene Angew. Math. 391 (1988), 85-99.

[Cp=l -, Duality in highest weight categories, Contemp. Math 82 (1989), 7-22. [CSP3] -, Integral andgraded quasi-hereditary algebras. I, J. Algebra 131 (1990), 126-160. [Cps41-, Abstract Kazhdan-Lusztig theories, Tohokn Math. J (to appear). [cm51 -, Infinitesimal Kazhdan-Lusztig theories, Contemp. Math. 139 (1992), 43-73. [Dl] V. Deodhar, Some characterizations of Bruhat ordering on a Coxeter group and determi-

nation of the relative Mobius function, Invent. Math. 39 (1977), 187-198.

PI -, On some geometric aspects of Bruhat orderings. II: The parabolic analog of Kazh- dart-Lusztig polynomials, J. Algebra 111 (1987), 483-506.

[Di] R. Dipper, Polynomial representations of finite general linear groups in non-decreasing characteristic, Prog. Math. 95 (1991), 343-370.

[DJ] R. Dipper and G. James, The q-Schur algebra, Proc. London Math. Sot. (3) 59 (1989), 23-50.

[DR] V. Dlab and C. Ringel, Quasi-hereditary algebras, Illinois J. Math. 33 (1989), 280-291. [GJ] 0. Gabber and A. Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Sci. Ecole Norm.

Sup. 14 (1981), 261-302.

c


[Gr] J. A. Green, Polynomials representations of GL, , Lecture Notes in Math., vol. 830, Springer-Verlag, Berlin and New York, 1980.

[H] D. Happel, Triangulated categories in the representation theory ofjinite dimensional algebras, London Math. Sot. Lecture Note Ser., vol. 119, Cambridge Univ. Press, 1988.

[Ho] J. Humphreys, Reflection groups and Coxeter groups, Cambridge Univ. Press, 1990. [Ill] L. Illusie, Complexe cotangent et deformations, Lecture Notes in Math., vol. 239, Springer-

Verlag, Berlin and New York, 1971. [Il] R. Irving, The soclefiltration of a Verma module, Ann. Sci. Bcole Norm. Sup. 21 (1988),

47-65. WI -, A jiltered category @’ , Mem. Amer. Math. Sot. 419 (1990). [Iv] B. Iversen, Cohomology of sheaves, Springer-Verlag, Berlin and New York, 1986.

[Jl J. C. Jantzen, Representations of algebraic groups, Academic Press, San Diego, 1987. [KLl] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, In-

vent. Math. 83 (1979), 165-184.

w4 -, Schubert varieties and Poincare duality, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Sot., Providence, RI, 1980, pp. 185-203.

[Ll] G. Lusztig, Some problems in the representation theory offrnite Chevalley groups, Proc. Sympos. Pure Math., vol. 37, Amer. Math. Sot., Providence, RI, 1980, pp. 313-317.

WI Hecke algebras and Jantzen’s generic decomposition patterns, Adv. Math. 37 (198oj, 121-164.

F.4 -, Modular representations and quantum groups, Contemp. Math. 82 (1989), 59-77. [LV] G. Lusztig and D. Vogan, Singularities of closures of K-orbits on a flag manifold, Invent.

Math. 47 (1987), 263-269.

PI B. Pat-shall, Finite dimensional algebras and algebraic groups, Contemp. Math. 82 (1989), 97-l 14.

[PSI B. Parshall and L. Scott, Derived categories, quasi-hereditary algebras, and algebraic groups, Carlton Univ. Math. Notes 3 (1988), l-104.

[PW] B. Parshall and J.-P. Wang, Quantum linear groups, Mem. Amer. Math. Sot. 439 (1991). [So] W. Soergel, n-Cohomology of simple highest weight modules on walls and purity, Invent.

Math. 98 (1989), 565-580. [Sp] T. A. Springer, Quelques applications de la cohomologie d’intersection, Bourbaki Sem.,

198 1 / 1982; Asterisque 92-93, 249-273. [Ve] D.-N. Verma, The role of afine Weyl groups in the representation theory of algebraic

Chevalley groups and their Lie algebras, Lie Groups and Their Representations, Wiley, New York, 1975, pp. 653-706.

[Vl] D. Vogan, Irreducible representations of semisimple Lie groups. II: The Kazhdan-Lusztig conjectures, Duke Math. J. 46 (1979), 805-859.

WI -, Representations of real reductive Lie groups, Birkhauser, Base1 and Boston, 198 1. [W] Jian-pan Wang, Partial orderings on a&e Weyl groups, J. East China Norm. Univ. Natur.

Sci. Ed., no. 4 (1987), 15-25. (Chinese) [Yj Jia Chen Ye, A theorem on the geometry of alcoves, Tongji Daxue Xuebao 14, no. 1

(1986), 57-64. (Chinese)

(EDWARD CLINE) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF OKLAHOMA, NORMAN, OKLAHOMA 73019

(BRIAN PARSHALL AND LEONARD SCOTT) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF VIRGINU, CHARLOTTESVILLE, VIRGINIA 22903-3 199

simulating perverse sheaves in modular representation...

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